Registering with the Police
If you want to do things 100 percent legally as a visitor to Montenegro, you can stop in at a police office and register. They give you a card with a stamp on it which you put inside your passport. So far I have never heard of anyone getting in any trouble from not having one of these cards and in fact I have been told by many Montengegrins not to worry about it. If you are staying in a registered hotel, the hotel will probably do this for you. I have been told that the reason to do this is in case they ask you where you have stayed while you were in the country, but so fr I have left the country several times and I haven't been asked for the card, and no one I know has been asked for it. (Note about Nathalia from Russia)
In Podgorica you go to the police station on the other side of the river from the center. It is about one city block past the Millenium bridge and you can only do this registration between 11:00 AM and 2:00pm. Before you go there, though, it is necessary to alreaday have the registration card with you. I have been told you get this in bookstores. They don't have them at the police office, so if you go there without the card they will just tell you to come bac when you have the card. By the way, when I have gone there were not many people who spoke English.
If you have more information on this, please write me.
The only person I personally know who was asked about this card was Nathalia from Russia. You can see her in the video about Marijana the artist, and you can see why she would attract attention! But Marijana was with Nathalia when the police officer stopped her on the streets of Kotor and Marijana told he police officer to stop hassling her! Still, just to be safe, Nathalia went to the police office later and registered, but this was very inconvenient because the police station is far from the Old Town of Kotor where she was questioned.
Additional Notes
One of my hostel guests from Slovenia was pretty worried about having this card when he crossed the border into Serbia, so he went over to the police office to get it. (He speaks Serbian so this was a bit easier for him.) I am guessing that he was more worried than other travellers because a few years ago they were more strict about this kind of thing in the former Yugoslavia countries. Of the 50 or so countries I have been to, I persponally have never had to go register with the police like this. I heard about something like this when you visit Russia, so I suppose it is some kind of an old communist style procedure. I encourage the Montenegrin Ministry of Tourism to change this to make it easier on both the tourists and the hotel owners. If you agree, please write to me and give me your thoughts, comments, experiences etc. I am also going to try to let people here know about Couchsurfing and how it operates so they will see this procedure doesn't fit well with this rapidly growing form of travel and accomodation.