Did granny have other kids who died? Just curious.

Ultimately, what granny did with her interest was her choice, and I'm afraid any discussion about what your mom would do with (formerly) granny's share wouldn't be enforceable. In other words, your mom didn't have to sign over half (?) interest to you in the first place, since she was her mom's only lawful heir (unless granny left you X in a will). Your mom somehow had the power to sign over half (?) interest in the place (I suspect mom and mom owned jointly w/ right of survivorship and your mom became sole owner when her mom passed ... and that's why your mom chose to sign over half interest to you and keep it JTWROS).

Whatever the carrying/maintenance costs are is likely far less than 50% of fair market value (let's say mom wants rent for her share), but you'd know better than we. You're free to unilaterally choose to "upgrade" the place, but any increase in value benefits both you and mom (you're free to point that out to her, with the help of a realtor or two's market assessment based on upgrades).

You don't "have" to do anything unless a court of law says you do (and you run out of appeals, if you lose).

"Can someone help me understand what my rights are?"

Your rights are to ignore her unless/until she does something (like inviting unsavory folks to come live there because you won't hand over some rent), hand over 50% of fair market value rent (or whatever's fair if you can agree), or offer (in writing) buy out her interest. You don't have to educate her on her options, but if you refuse to buy her out or put place up for sale, her recourse legally would be to file a partition action and ask a court to order it sold.