You have hit on an interesting (read: highly annoying) bug in the App Store validation process, it seems:
For starters, the location to which you copy your apps does not, in fact, come into it, and neither does your status as admin / non-admin user. Despite the fact Apple states this should be possible, on my two machines, I cannot get an app manually copied to run either as a standard user, an admin user, in /Applications or in ~/Applications. Neither does chowning and chmoding the bundle to conform with regular installs help. The App Store login sheet pop ups (once), but the app keeps quitting with exit code 173 after that.
According to Apple’s developer documentation for the validation of App Store receipts, part of the receipt validating the use of an app is a machine specific GUID. When you copy an app to another machine, the GUID does not match anymore, thus invalidating the receipt (conversely, on your own machine, it always matches, whatever the user you install the app for – hence you not having any problem with that). Apps are meant to signal an invalid receipt by exiting with exit code 173, and the App Store’s storeagent process is meant to take over after that, asking for credentials and generating a new receipt, then re-launching the app on success:
If the system successfully obtains a valid receipt, it relaunches the application. Otherwise, it displays an error message to the user, explaining the problem.
Do not display any error message to the user if validation fails. The system is responsible for trying to obtain a valid receipt or informing the user that the receipt is not valid.
What happen in your (and my test) case is that validation fails quietly – as indicated by the fact the receipt is never updated, but no error message is shown either. The logs show storeagent returns:
promptResponse: <CKSignInPromptResponse:0x1080c4fb0 returnCode:1>
This code is undocumented by Apple, and all I have found are discussions by developers about this issue in a testing context, where it seems to relate to iTunes Connect data being incomplete (Geoff Pado has a concise summary here).
However that may be, the picture as a whole clearly shows validation is failing en route, which leaves apps copied manually with an invalid receipt even after entering the correct credentials to the App Store. Only those apps that do not check their own validity will still run, as you noted.
Not very helpfully in your case, Apple only recommends deleting and re-downloading from the App Store to remedy the issue.
TL;DR: You cannot manually copy apps bought in the App Store from one computer to another until Apple fixes the receipt validation process needed for it. Install location and user privilege do not come into it.
Best Answer
You may have an application which was left over from the previous owner. The simple way to deal with that is to delete the old version and download the new version (with your own Apple ID) anew.
First, in the Mac App Store, go to the Purchases tab and verify that the app you want to download is listed there for your Apple ID.
Next, move (or, if you are brave, remove) the application from the /Applications/ folder to someplace like your Desktop.
Finally, quit/restart the Mac App Store, go back to the Purchases tab, and re-download the application.