When I became a landowner I got some shocking--but good--advice on the Attractive Nuisance Doctrine: I was now legally liable for the safety of careless children, trespassers, vandals, and even thieves. I was expected to take precautions for dogs and cats who might come sniffing around where they didn't belong.
The fact is that the law protects people (especially children) and common courtesy protects dogs (and cats) on private property where they don't have any right to be. The common courtesy is written into the MTA Trapping Ethics on avoiding the use of non-specific lures and lethal traps where they'd be likely to attract and take domestic animals.
Now, like all of my neighbors, I don't leave a hole uncovered, a sharp-edged tool out, or a machine startable. If I need a permit, I fill out the form, draw the plan, and answer the questions. If I had to deal with a pest-control problem, I would only use a trapping method that had an acceptable safety level for people and domestic animals. People and pets are safe on my land in the open space to the south, the shortcut to the north, and (although I might give them a hard time about it) anywhere in-between. The law and ethics are clear: I'm responsible for any dangerous structures or equipment I might leave on my private property.
Some things depend on the circumstances. If there was a locked fence around the place, a trap in the yard wouldn't be considered so dangerous. If the trap was covered so that the only entry was underground, most of my neighbors wouldn't consider it dangerous at all. They'd consider a trap in the garage or house to be the most important and the least dangerous of all.
When I ask for reasonable trapping regulations on private property, I make allowances for fenced farmland, large rural properties with buffer zones, and other areas determined necessary by responsible officials and wildlife-management personnel. This doesn't always make the best sound byte or the simplest plan, but it addresses more of the complicated problem of responsible wildlife management than simpler plans.
So here's the question: should trapping be regulated on private property? I say yes, with allowances for the circumstances. In my residential community, I'm asking for Ordinances that ensure the safety of people and domestic animals anywhere in the community. On a fenced, rural farm, I'm not asking for those Ordinances--because I hear that the safety concerns are different there. Statewide, I hear that private property falls under many different categories, so my ideal would be to talk to the experts about making appropriate allowances for each category. And then when the experts told me that I couldn't have such a complicated system, I'd ask for a single set of allowances for rural private property, with a 100-yard buffer zone where it adjoins property without those allowances (i.e. where it adjoins public property or where it adjoins private property where the owner doesn't take the allowances).
See also: trappers' double standard on private-property rights.