If two tenants are renting a suite together as "co-tenants" and one tenant wants to give notice and the other doesn't want to - how does that work? What rights does the tenant who wants to leave have?

check your contract.

i don't know where you live, but i live in a shared house and if someone wants to leave, they are able to; but they have to find a replacement tenant and give a months notice (as we pay rent monthly.)

in my contract it actually says that you can leave providing "you find a suitable tenant, both to the agency and also to the remaining tenants."

Depends where you are in the contract. If its a six month contract, and you are on month six, you can give your notice and leave no problem- if the other person wants to extend the contract, its their problem to find someone new to do it with. If you, however, on say month 3, you will need to find a "suitable replacement" and get them, together with your old co-tenant and landlord, to sign a modification of the existing contract.

There's a rental agreement, but no term-based contract. It's month to month. I'm the one who wants to leave, essentially. My concern is with the security deposit. Can I just go to the landlord and say I'm leaving, I want my security deposit back and then he hassles her to replace it? What if she refuses?

Well if she refuses, then he'll make her move out. Thats life- you've either got to pay the full security deposit or you can't live somewhere. And if someone was living there and chose to leave, taking their half with them, you either replace it or go.