What is the correct way of visiting Transnistria without violating Moldavian laws?

Visiting Transnistria should not be a problem, both with entering it and leaving it from Ukraine and Moldova.

Transnistria controls the territory, and Moldova considers the territory its own part. This effectively means that Transnistria-Ukraine border is not controlled by Moldova even though Moldova treats tourists as visiting their territory. On the Transnistria-Moldova border they don't have the border officials because it's "internal" border, however they do have customs there so that one could do customs formalities because it's the last point on the route through Transnistria where it is at all possible.

I visited Transnistria twice with entry/leave from Moldovan side in 2015 on a Ukrainian passport. Things to be aware of:

• in Transnistria one has to register within 1 day. If you don't leave the same day then you have to be aware of it. I would expect questions of border service to those who cannot provide proof of registration (here, however, it is mentioned that if one puts certain address (hotel or relative) in a form on entry, then one has 45 days of registration free stay, however it is limited to CIS countries passport holders and other still have 1 day, or even 10 hours; I would double check that on the border because I was asked if I am going to leave the same day or not)
• if you are going by car:
  • Green card liability insurance is not valid in Transnistria and they sell their own mandatory insurance right on the border
  • if you are going along the route Moldova-Transnistria-Ukraine, you typically purchase vignette (road tax) when entering Moldova. The vignette has expiry date and leaving through Transnistria you give no chance to Moldovan officials to see when exactly you left and that you left at all. Next entry to Moldova might be a surprise and a question about your possibly driving Moldovan roads without valid vignette all this time... To get yourself out of trouble, when crossing Moldova-Transnistria border make sure to find customs officer (it might so happen they don't stop you and you drive pass-through; also they might be located in a weird box by the road without any descriptive title on it) and cancel the vignette or otherwise obtain a proof of your leave.

To my best knowledge Transnistria officials do not stamp passports. Instead you fill a form when you enter, then are supposed to pass its stamped part back when you leave Transnistria. There is no really a problem that getting stamped one might get problems elsewhere in future.