I would like to visit Transnistria this summer. Assuming I'm allowed into Transnistria visa-free, do I have to get myself registered upon arrival? If so, how does the procedure work?
By reading this link, it doesn't appear that you need to register with the police.
The procedure for entry into Transnistria from Moldova is pretty straightforward (as of March 2015); any travel document valid for Moldova and/or Ukraine (depending on point of entry) is sufficient. You will be asked by the border guards (some of them speak decent english) your reason for visit (tourism/business), and what hotel you plan to stay at. Then you will be given a entry card that allow you to stay in Transnistria for 24 hours. Be sure not to lose this important piece of paper, and above all, make sure you're actually checked and get it, because, at least when entering by minibus, guards often do not perform checks (though they definitely will on exit) and not having an entry card is almost guaranteed to lead to being asked for a bribe, see below
If you want to stay longer, you will have to register at your hotel, the Ministy of Interior or the immigration office in Tiraspol. You only need your passport, the piece of paper from the border and a receipt from your hotel. Keep the small square piece of paper they give you and present it to the guards when you leave. It's all simple, free and easy.
So, it seems you need to register with your hotel and the Immigration Office. All I can found info wise on the police is about corruption, check the link for details.