Am I allowed to install Snow Leopard on my MBP using someone elses install disc?
Yes, if it's an install disc that was either a boxed copy, or an up-to-date CD, no if it was just the disc that came with the machine unless the model number of the machine is the exact same one.

There are no license keys or otherwise.

You may not be able to install bundled versions of iLife as they are often locked to the machine it was bought from (but even this is get-roundable by showing package contents and using the individual packages rather than the wrapper installer), but the OS is usually fine.

I think the only install media that isn't usable across a number of systems are the Lion USB installers which I believe are tied down at serial number granularity.

Frankly, if you are worried about licensing, just upgrade to Lion for $30.