NEVER hand out keys. You're correct in that keys would be obtained through the Owner/management. Certainly you should check your lease to see if anything pertains to showing if you don't renew but that is usually based on showing your legally rented apartment at a time that is convenient for you. Demand that any perspective tenants not view alone but are in the company of a licensed real estate pro. We are required by NY State to carry our pocket licenses and we are also required to show it on demand. I would ask to see it before allowing anyone in and if the agent doesn't have it, I would NOT allow them access. Not sure if the info you seek is in the NYS Tenants Guide so, here's the link:
I represent owners with tenants in apartment. In some cases all communication for showing requests is through owner. Owner contacts tenant for showing request and then confirms with broker. I may contact tenants direct when given permission by both landlord and tenant to notify them 24 hours in advance of a requested showing.
Usually permission to enter is through tenant but landlord must notify tenant that I am the broker and will be contacting tenant. I would not give your keys to anyone that you do not know and your landlord did not tell you to.
Since you or your roommate works at home only schedule showings when one of you are home. You can either be present or if you don't want to be in the apartment while it's being shown just open door and walk around the block for 10 minutes or where ever.
There is also someone here every day to allow access since one of us works from home so access is not an issue. According to the lease it says can show with reasonable notice.
According to my insurance company, I could be liable for theft or damage if keys are supplied to people other than the tenant or landlord.