I see a lot of people in this thread state unjustified assumptions as if they are proven facts. I'll try and comment on a few of these, judging from my experience with how things transpired here.
* This will make it a shelter for the homeless *
There have always been homeless people on the busses, and I didn't notice a markedly increase. The fact that you still need a legitimate ID card and a fare card might help this point.
* This will multiply users of the buses, and lead to frustration and long queue lines *
Utilisation of public transportation increased by 15% since becoming free. I noticed somewhat more people, but nothing annoyingly so. On the other hand, the main goal as I could see was to get cars off the road. A drop in traffic of 14% was observed.
Because this is a low-wage country, the €12 million in lost revenue this cost is quite low in absolute terms, compared to what it would be in more western, bigger cities, but it's still a large chunk of money relative to GDP here. Due to the requirement that you need to be a resident of this city, 10000 more people have "migrated" to Tallinn from other cities, now paying taxes here, and adding a predicted €9 million in tax revenue for Tallinn.
Personally I love it of course. I use it in ways that others might deem "uneconomically" i.e. take the tram for 2 stops instead of walking 15 minutes, but I also notice I travel further distance more often, exploring the city, because I don't feel annoyed with having to either find parking space, or pay a taxi, or pay a bus fare.