Visiting And Shopping Inside And Outside The Former Soviet Union.

[Al-muell]

That process you are talking about is called "registration", and is common thorughout the not-so-free former republics (basically all but the Baltics).

My experience with Ukraine (which IS not-so-free) was that You had to fill in this form, but then You would give half of it to the customs official at the border and the second half you would keep in Your passport on event of police control, but no stamp was neccessary there. But maybe because Poland had separate contract (e.g. no additional visa required) with Ukraine it was different.

As for bribery - In Russia the cops are paid silly money, so they look for every occasion. However, it doesn't apply to all the cops - Road "menti" seek for any opportunity to earn extra $$$, while most of the criminal investigation policemen would not hesitate to arrest you when proposed a bribe. It's less of a problem in Ukraine, and from what I know You shouldn't try to bribe ANYONE in Belarus.
Also, not everyone in uniform who wants (and ussually has a right) to check Your ID is a policeman.