As a new CFI, here is what I have learned while researching aircraft insurance for CFIs.
Most insurance policies are written to only cover the "named insured", in this case your student (owner of the airplane). As a CFI, you can request to be added onto the policy as a named and approved pilot. This would mean that the insurance coverage is extended to you (liability for bodily/property and aircraft damage). A trap I have heard is some policies will not allow a named and approved pilot if they are conducting a commercial operation (flight instruction included). I believe the best thing to do in this situation is call the company and have a written record that you are added to the policy and that the coverage extends to flight instruction.
The second thing I have run across is a "waiver of subrogation". Depending on the aircraft owner's policy, the insurance company usually reserves the right to reclaim any losses. In short, this means they may come after you if you cause any damage/claims, even if you are a named and approved pilot. (Google "subrogation"). Some companies will issue a "waiver of subrogation" meaning they forfeit their right to try and recover from you.
This article from NAFI is pretty helpful. Some CFIs also opt for their own independent non-owned liability coverage for bodily/property and aircraft damage.
What You Don't Know Can Hurt You!
And I am by no means an insurance expert, just a CFI who has done some reading! I would say consult with an insurance expert before deciding what you do.