Pretty much as Paatti said.
The number of bears is growing so much that the Ministry of Agriculture and Forestry is actually considering to increase the number of licences by 40 this year.
Eastern and Northern Finland are the prime area, although also in Central Finland there is nowadays a fairly healthy bear population.

I don't have statistics on the guns, but the .308, .30-06 and 7,62x53R seem to be prevalent.
More experienced bear hunters have gone for the 9,3x62 or .45-70 lever guns.
I doubt if any sane person would use a semi-auto for dangerous game like bear. The bolt actions would be the most commonplace.
This year I have one bear licence for Lapland. I'm torn between my 8,2x53R lever action and my father's more potent 9,3x62 bolt action which I'm less familiar with.

A dog is considered to be fairly essential in bear hunting.
However, more experienced bear hunters have told me that few dogs, indeed, have the courage to go against bear.
One of them told an amusing story about a "tough" police dog who came yelping with its tail between its legs and took cover behind the police once it got the scent of the bear tracks!
Karelian bear dogs are often used, and the Russian laikas are considered good bear dogs, as well.
Apparently, the most common technique is that the dog will "stop" the bear by barking thus giving the hunter the opportunity to approach and shoot.

I believe some bears are also taken from stands where you wait in the dawn for the bears to come and feed e.g. on fields. As Paatti mentioned, no baiting, though.
At least this is commonplace in Karelia on the Russian side of the border.

- Lars/Finland

PS. Jordan: Thank you very much for the excellent post!