A tentant can have their own renters insurance which will insure their contents and provide them with some liability insurance. Additionally there would be coverage is someone slips and falls and miscellanous coverage.
They can insure their contents, but I don't think they can insure the property. I can't remember what they call it, but there is some kind of clause I believe in the policies that says something like an uninterested party can not benefit from the policy. That's not the right term, but its something like that.
Yes, they insure the contents and the owner continues to insure the property -- unless the tenants is renting furnished, in which case, as an owner, be sure you have sufficient coverage for the contents as well.