Do GNU/Linux programs usually check local directory for libraries?

Slightly related to what I was saying here, this is a straightforward question.

I didn't compile very many programs myself so I'm wondering. How common is it for a program, once you compile it and have the binaries ready in the directory in which you ran "make", to check for libraries and dependencies in the current local path and not just paths like /usr/bin and /usr/libs.

The reason I am asking is because I have this idea of perhaps switching completely to Debian Stable, therefore getting a rock solid system, but still allowing myself to run the latest applications available from original developers by using something like Debian Backports OR, even better perhaps, compile my own software, put the resulting binaries and all their dependencies in a single archive and then uploading it for others to use.

People could then basically install those archives on pretty much any distribution - just decompress to /programs and run. I could make a script to add shortcuts to menus and/or /home/user/Desktop or a special shortcuts folder for easy access.

And if it works it could be a start of a new universal package format for ALL GNU/Linux. We could call it bin.tar.gz or something. Eventually we could even join together to build a whole repository of these kinds of packages which will work everywhere and have a special program for checking, fetching and installing them.

It's like source tarballs, except that they're binary tarballs.