Maybe these schema can clarify the situation. This is the usual setting:

                               Terminal  (/dev/ttyX or /dev/pts/x)
                                device
                                   |
                    (screen)<--[<output]----x-------(stdout) Process1
        Terminal  (keyboard)---[input >]---o-\----->(stdin)
                                            \ \
(hardware console or                         \ `----(stdout) Process2
 virtual console or terminal                  `---->(stdin)
 emulators like xterm, …)

And there is no way to plug some new Process3 like this:

                             Terminal
                              device
                                 |
             (screen)<---o---[<output]--x------(stdout) Process1
  Terminal (keyboard)---/-x--[input >]-o-\---->(stdin)
                       | /              \ \
                       | |               \ `---(stdout) Process2
                       | |                `--->(stdin)
                       | |
                       \ `---------------------(stdout) Process3
                        `--------------------->(stdin)

What screen (and others) does is allocating some pseudo terminal device (like xterm does) and redirect it to one or more "real" terminals (physical, virtual, or emulated):

             Terminal                   pseudo
             devices              ,--> Terminal (/dev/pts/x)
                |         _______/      device
Terminal <--[<output]--- |       |        |
 1       ---[input >]--> |screen | <--[<output]---x-----(stdout) Process1
                         |Process| ---[input >]--o-\--->(stdin)
Terminal <--[<output]--- |       |                \ \
 2       ---[input >]--> |_______|                 \ `--(stdout) Process2
                                                    `-->(stdin)

Using screen -x you can attach one more terminal, xterm, whatever (say Terminal 3) to the screen session.

So no, you can't communicate directly through stdin/stdout with processes attached to a different terminal. You can only do so through the process that is controlling this terminal if it happens to be a pseudo terminal, and if this process was concieved to do so (like screen is).

Terminal emulators

The master side replaces the line (the pair of TX/RX wires) that goes to the terminal.

The terminal displays the characters that it receives on one of the wires (some of those are control characters and make it do things like move the cursor, change colour...) and sends on another wire the characters corresponding to the keys you type.

Terminal emulators like xterm are not different except that instead of sending and receiving characters on wires, they read and write characters on their file descriptor to the master side. Once they've spawned the slave terminal, and started your shell on that, they no longer touch that. In addition to emulating the pair of wire, xterm can also change some of the line discipline properties via that file descriptor to the master side. For instance, they can update the size attributes so a SIGWINCH be sent to the applications that interact with the slave pty to notify them of a changed size.

Other than that, there is little intelligence in the terminal/terminal emulator.

What you write to a terminal device (like the pty slave) is what you mean to be displayed there, what you read from it is what you have typed there, so it does not make sense for the terminal emulator to read or write to that. They are the ones at the other end.

The tty line discipline

A lot of the intelligence is in the tty line discipline. The line discipline is a software module (residing in the driver, in the kernel) pushed on top of a serial/pty device that sits between that device and the line/wire (the master side for a pty).

A serial line can have a terminal at the other end, but also a mouse or another computer for networking. You can attach a SLIP line discipline for instance to get a network interface on top of a serial device (or pty device), or you can have a tty line discipline. The tty line discipline is the default line discipline at least on Linux for serial and pty devices. On Linux, you can change the line discipline with ldattach.

You can see the effect of disabling the tty line discipline by issuing stty raw -echo (note that the bash prompt or other interactive applications like vi set the terminal in the exact mode they need, so you want to use a dumb application like cat to experience with that). Then, everything that is written to the slave terminal device makes it immediately to the master side for xterm to read, and every character written by xterm to the master side is immediately available for reading from the slave device.

The line discipline is where the terminal device internal line editor is implemented. For instance with stty icanon echo (as is the default), when you type a, xterm writes a to the master, then the line discipline echoes it back (makes a a available for reading by xterm for display), but does not make anything available for reading on the slave side. Then if you type backspace, xterm sends a ^? or ^H character, the line discipline (as that ^? or ^H corresponds to the erase line discipline setting) sends back on the master a ^H, space and ^H for xterm to erase the a you've just typed on its screen and still doesn't send anything to the application reading from the slave side, it just updates its internal line editor buffer to remove that a you've typed before.

Then when you press Enter, xterm sends ^M (CR), which the line discipline converts on input to a ^J (LF), and sends what you've entered so far for reading on the slave side (an application reading on /dev/pts/x will receive what you've typed including the LF, but not the a since you've deleted it), while on the master side, it sends a CR and LF to move the cursor to the next line and the start of the screen.

The line discipline is also responsible for sending the SIGINT signal to the foreground process group of the terminal when it receives a ^C character on the master side etc.

Many interactive terminal applications disable most of the features of that line discipline to implement it themselves. But in any case, beware that the terminal (xterm) has little involvement in that (except displaying what it's told to display).

And there can be only one session per process and per terminal device. A session can have a controlling terminal attached to it but does not have to (all sessions start without a terminal until they open one). xterm, in the process that it forks to execute your shell will typically create a new session (and therefore detach from the terminal where you launched xterm from if any), open the new /dev/pts/x it has spawned, by that attaching that terminal device to the new session. It will then execute your shell in that process, so your shell will become the session leader. Your shell or any interactive shell in that session will typically juggle with process groups and tcsetpgrp(), to set the foreground and background jobs for that terminal.

As to what informations are stored by a terminal device with a tty discipline (serial or pty), that's typically what the stty command displays and modifies. All the discipline configuration: terminal screen size, local, input output flags, settings for special characters (like ^C, ^Z...), input and output speed (not relevant for ptys). That corresponds to the tcgetattr()/tcsetattr() functions which on Linux map to the TCGETS/TCSETS ioctls, and TIOCGWINSZ/TIOCSWINSZ for the screen size. You may argue that the current foreground process group is another information stored in the terminal device (tcsetpgrp()/tcgetpgrp(), TIOC{G,S}PGRP ioctls), or the current input or output buffer.

Note that that screen size information stored in the terminal device may not reflect the reality. The terminal emulator will typically set it (via the same ioctl on the master size) when its window is resized, but it can get out of sync if an application calls the ioctl on the slave side or when the resize is not transmitted (in case of an ssh connection which implies another pty spawned by sshd if ssh ignores the SIGWINCH for instance). Some terminals can also be queried their size via escape sequences, so an application can query it that way, and update the line discipline with that information.

For more details, you can have a look at the termios and tty_ioctl man pages on Debian for instance.

To play with other line disciplines:

1. Emulate a mouse with a pseudo-terminal:

   socat pty,link=mouse fifo:fifo
   sudo inputattach -msc mouse # sets the MOUSE line discipline and specifies protocol
   xinput list # see the new mouse there
   exec 3<> fifo
   printf '\207\12\0' >&3 # moves the cursor 10 pixels to the right
   

   Above, the master side of the pty is terminated by socat onto a named pipe (fifo). We connect that fifo to a process (the shell) that writes 0x87 0x0a 0x00 which in the mouse systems protocol means no button pressed, delta(x,y) = (10,0). Here, we (the shell) are not emulating a terminal, but a mouse, the 3 bytes we send are not to be read (potentially transformed) by an application from the terminal device (mouse above which is a symlink made by socat to some /dev/pts/x device), but are to be interpreted as a mouse input event.

2. Create a SLIP interface:

   # on hostA
   socat tcp-listen:12345,reuseaddr pty,link=interface
   # after connection from hostB:
   sudo ldattach SLIP interface
   ifconfig -a # see the new interface there
   sudo ifconfig sl0 192.168.123.1/24
   
   # on hostB
   socat -v -x pty,link=interface tcp:hostA:12345
   sudo ldattach SLIP interface
   sudo ifconfig sl0 192.168.123.2/24
   ping 192.168.123.1 # see the packets on socat output
   

   Above, the serial wire is emulated by socat as a TCP socket in-between hostA and hostB. The SLIP line discipline interprets those bytes exchanged over that virtual line as SLIP encapsulated IP packets for delivery on the sl0 interface.