Any way to make a binary in a Linux distribution and run it on another distribution with same architecture? Or I should compile and build it on different distributions?
Statically linking your binaries makes them LESS portable because some libraries won't then work correctly for that machine (differing authentication methods, etc).
If you statically link any "unusual" libraries and keep your set of supported distros to a minimum, you should be ok.
Don't statically link the C library (or the whole binary), that's a recipe for trouble :)
Look at what (e.g.) Google do with Chrome.