As I read the question, the contract is with and the rents are paid to the landlord by the Sister. I am unable to find in the 1099 instructions where it says that an individual tenant must issue a 1099 for rents paid.
If the Motherhouse is paying the Sister a housing stipend, I could see where the Motherhouse might want to issue a 1099 to the Sister.

There's enough ambiguity in the OP to drive a truck through. You are correct that an individual tenant does not issue a 1099 (unless the rent is paid in connection with a business, but that's not the case here). I'm not conversant in the financial operations of all religious entities, but if the rent is paid out of a communal clergy fund, then the 1099 might be appropriate.

As to your final statement, if the Motherhouse is paying the Sister a housing stipend, that would be reported on a W-2 as a parsonage allowance (box 14).

Ira