Here in Ohio I believe the most recent info I read was that both landowners are responsible for the fence, half and half like said before. The state Farm Bureau should have resources handy on this issue. I have never dealt with damages before, though.
Just an added note on building the fence back off the property line (and of course, in this case when it is completely on your property all the expense is yours), a permanent fence should be recorded as to the location and distance from the surveyed boundary at the county recorder, cross-referencing on your deed. If not, the next surveyor may just follow the fence line and GPS is not accurate under tree cover.
DH has driven steel T-posts along all the boundaries even though our temp-type fences aren't on the line. Our grown kids know about this for future reference. This way he can mow both sides, although the goats take care of the brush on both sides in their field. They know when we don't have the electric on it. There is just vacant land adjacent, but our first pair of horses and mule got scared one night (bear?) and took off running, downed 4 strands of barbed wire (they were in a rather small paddock area at the time next to the woods), and we met our "neighbors" at 0200! It was scary because they had gotten down to the highway 1 1/2 miles away. They woke the dogs, the dogs woke the valley up and someone got our number...how embarrassing.
I think it helps to have tame livestock too. The girls I have now seem to want to come to the barn or house if they get out. I have awaken to find my two mares and my cow in the rose garden!