I think how much the postdoc should be independent and how much she shares the result with you depend on the source of money, and written or non-written mutual agreement. If her salary comes from your research money, and the contract says she is a part of your (bigger) project, she has a duty to spend her time (at lease some reasonable amount of time) to accomplish the goal of the project and report/discuss result to/with you. If she is paid by her own grant/scholarship or is paid by university, she may have more freedom on her project. But it'd be still beneficial for her to work with you and collaborate (and share result with you by co-authering). However, what makes me puzzled is that she doesn't want to discuss with you and share the result (and authorship) with you: If this is always the case, there is no reason for you to hire (whether you pay or not) her. Even if you don't pay her, if she doesn't collaborate with you nor shares, you're wasting your time and resource which you might have used to collaborate with others otherwise.

Assuming her goal is to become a professor or independent researcher like you, she might be in a pressure of proving herself through publication and research independence. She might expect also an authority equivalent to yours even though it's a wrong expectation for her position as postdoc.
I think it's worth addressing these issues and be on the same page as far as expectations before you fire her. Whether it's personal relationship or leadership, it's very important to synchronize the expectations and communicate. Without being emotionally hurt, I believe you can point out the expectation discrepancies between you and your friend and be clear that you can help her to achieve her goal (e.g. by helping her to get mentored grant for postdocs) and give her independence which amount reasonable to both of you, and in return you expect that she will share the result (and authorship) on her independent projects (you deserve it as a give-and-take since you help her to keep the affiliation, provide resource and help her to grow) as well as contribute/collaborate on your projects. I'd make the argument more win-win than threatening her. It's also a good idea discussing with your colleague and department before and while you talk with her- you may approach them to get some leadership/communication advice on your situation rather than to accuse her. Keep them posted while you talk with your friend (and document the procedure/evaluations/key conversations if possible). When you have gone through all reasonable efforts and she still doesn't get it, you have grounds and witnesses for firing her, so don't need to worry about a backfire. It's easier said than done, but try to be proactive and win-win. Good luck.

I think how much the postdoc should be independent and how much she shares the result with you depend on the source of money, and written or non-written mutual agreement. If her salary comes from your research money, and the contract says she is a part of your (bigger) project, she has a duty to spend her time (at lease some reasonable amount of time) to accomplish the goal of the project and report/discuss result to/with you. If she is paid by her own grant/scholarship or is paid by university, she may have more freedom on her project. But it'd be still beneficial for her to work with you and collaborate (and share result with you by co-authering). However, what makes me puzzled is that she doesn't want to discuss with you and share the result (and authorship) with you: If this is always the case, there is no reason for you to hire (whether you pay or not) her. Even if you don't pay her, if she doesn't collaborate with you nor shares, you're wasting your time and resource which you might have used to collaborate with others otherwise.

Assuming her goal is to become a professor or independent researcher like you, she might be in a pressure of proving herself through publication and research independence. She might expect also an authority equivalent to yours even though it's a wrong expectation for her position as postdoc.
I think it's worth addressing these issues and be on the same page as far as expectations before you fire her. Whether it's personal relationship or leadership, it's very important to synchronize the expectations and communicate. Without being emotionally hurt, I believe you can point out the expectation discrepancies between you and your friend and be clear about that you can help her to achieve her goal (e.g. by helping her to get mentored grant for postdocs) and give her independence which amount reasonable to both of you, and in return you expect that she will share the result (and authorship) on her independent projects (you deserve it as a give-and-take since you help her to keep the affiliation, provide resource and help her to grow) as well as contribute/collaborate on your projects. I'd make the argument more win-win than threatening her. It's also a good idea discussing with your colleague and department before and while you talk with her- you may approach them to get some leadership/communication advice on your situation rather than to accuse her. Keep them posted while you talk with your friend (and document the procedure/evaluations/key conversations if possible). When you have gone through all reasonable efforts and she still doesn't get it, you have grounds and witnesses for firing her, so don't need to worry about a backfire. It's easier said than done, but try to be proactive and win-win. Good luck.