Find -exec {} + argument list limit

You don't need any of the GNUisms here (and you probably want a -mindepth 1 to exclude .), and you don't need to run one chmod per file:

find . ! -name . -prune ! -type l -size +100c -size -1000c -print \
  -exec chmod a+r {} + >testfile

(I've also added a ! -type l because -size would check the size of the symlink while chmod will change the permissions of the target of the symlink so it doesn't make sense to consider symlinks. Chances are you'd want to go further and only consider regular files (-type f))

That works here because chmod doesn't output anything on its stdout (which otherwise would end-up in testfile).

More generally, to avoid that, you'd need to do:

find . ! -name . -prune ! -type l -size +100c -size -1000c -print -exec sh -c '
  exec cmd-that-may-write-to-stdout "$@" >&3 3>&-' sh {} + 3>&1 > testfile 

So that find's stdout goes to testfile but cmd-that-may-write-to-stdout's stdout goes to the original stdout before redirection (as saved with 3>&1 above).

Note that in your:

find . -maxdepth 1 -size +100c -size -1000c -exec chmod a+r {} \; -print > testfile

testfile would contain the files for which chmod has succeeded (the -print being after -exec means -exec is another condition for that -print, and -exec succeeds if the executed command returns with a non-zero exit status).

If you wanted to use xargs (here using GNU syntax), you could use tee and process substitution:

 find . ! -name . -prune ! -type l -size +100c -size -1000c -print0 |
   tee >(tr '\0' '\n' > testfile) |
   xargs -r0 chmod a+r

To save the output of find with NULs turned into newlines into testfile. Note however that that tr command is running in background. Your shell will wait for xargs (at least, most shells will also wait for tee and find), but not for tr. So there's a little chance that tr has finished writing data to testfile by the time the shell runs the next command. If it's more important that the testfile be fully written by then than all the permissions be modified, you may want to swap the xargs and tr commands above.

Another options is to wrap the whole code above in:

(<that-code>) 3>&1 | cat

That way, the shell will wait for cat and that cat will only exit when all the processes that have that file descriptor 3 open on the writing end of the pipe it reads (which includes tr, find, tee, xargs) have exited.

Another option is to use zsh globs here:

files=(./*(L+100L-1000^@))
chmod a+r $files
print -rl $files > testfile

Though you could run into a too many arguments errors if the list of files is very big. find -exec + and xargs work around that by running several chmod commands if needed. You can use zargs in zsh for that.