this was told by a table mate on a cruise a few yrs back.
seems they rented a car in cozumel to do a tour on their own. somewhere along the way a police man stopped them for speeding, which they didnt belive they were speeding but did not argue with the officier and the police told them they must go to the police station, to follow him, which they did, after a few miles the police officer pulled over and told them to keep following him, which they did. this happened 2 or 3 times. finally the police officier told them they could just go on their way, but not to speed again.
they headed right straight back to the cruise ship ,believing the officier wanted money, a bribe which he did not get.
I believe cozumel to be a safe port, but I did believe these people when they told it.