Whose responsibility is to setup one-on-one meetings? Is it the manager or the employee? What about performance reviews? My manager just sent a schedule for performance review and said I will work with you but did not setup any meeting. Does this mean I have to schedule one?
Who should be driving performance discussions? The manager or the employee?
The manager. As the name implies, managing people should be the most imporant part of the job and performance discussions and reviews are some of the most important responsibilities in people management.
It should be up to the manager to give consistent and regular feedback to the employees that they manage. Managing is all about building a strong team that can deliver results. A manager needs to coach his reports so that those who are struggling have a realistic path towards improvement but also so the ones that are doing well are given what they need to do even better.
All this means that it's the manager's responsibility to provide feedback moments to their employees in whatever way that they prefer. Typically the company has procedures on this but each manager will have a different way of giving feedback and preparing for formal reviews.
The actual scheduling itself is a simpler matter. That's more of an admin task so managers will routinely ask their employees to schedule those feedback moments themselves. The manager will give some indication of how often to meet or when specific formal steps need to be done. Some will prefer to schedule things themselves, some will just ask you to do so.
In your case, it sounds like your manager didn't tell you who should schedule the actual feedback conversation. So just ask them.
Of course this assumes a semi-competent manager and there are plenty out there who aren't. In case the issue is less about scheduling and more about an overall lack of feedback or a manager who's not taking performance meetings seriously, you need a different approach. While it should be up to the manager to give good feedback and have performance discussions, if they don't that doesn't mean you can't push for that! And you should. You can set-up additional feedback comments, ask for actionable feedback, discuss ways to improve and so on.
As an example, plenty of managers only do the bare minimum of feedback conversations that are required by company procedures. But that doesn't mean you can't set-up additional conversations outside that formal process. You get to push for feedback and input from your manager. It may take more work from your side than it should but it's absolutely worth doing. Even bad managers shouldn't object to you asking for more feedback.
Generally the person who wants the meeting sets it up
As a general rule, it makes more sense for the person requesting it to set it up as they are the one who may need to prepare for the meeting. If you are seeking a raise and want to prepare a case for that, scheduling the meeting yourself could make sense.
If you want to have this meeting, prompt your manager to schedule it or schedule yourself. Who does it exactly in that case is irrelevant. If you don't want it, you don't need to schedule it and there is a decent chance it will be forgotten. I find virtually any kind of meeting to not be terribly useful, so am usually happy to let ambiguity lead to not having a meeting. 80% of the time, the hypothetical meeting is never mentioned again and the other 20% of the time, they schedule it.
Whose responsibility is to setup one on one? Is it the manager or the employee?
It's just a meeting. I wouldn't overthink it. I would just ask my manager if they would like to set up the 1:1 meeting or if they would like me to set one up. Usually I would say the onus is on the manager to set up the performance 1:1 with their direct report. But if you have regular 1:1s with other people or if you have skip level 1:1s, then I would say the onus is on you.
The responsibility for 1:1s and performance reviews are not always the same.
Performance reviews, whether held quarterly or annually are the managers responsibility to ensure these meetings take place. This does not necessarily mean that they will own the calendar appointment, they could for example tell you to book a free slot in their calendar but if you do not then they should follow this up with you directly.
1:1 checkups for building working relationships, development, coaching, etc. are not so straight forward. The ownership will depend upon organisation and the individuals involved. Personally, I send my directs a list of available times with the request to select a slot that works for them and I create the booking, but if you let the report own the meeting this has the benefits of the manager needing to consult the report to change the calendar item.
If your manager has sent you a list of available times (or a schedule) for your performance review, it is possible they are simply waiting for a response with a time that suits you. It's impossible to say without seeing the email so as the other answers mention, rather than waiting and wondering who is responsible, please ask them. Being proactive and showing initiative when communication breaks down is what you should do in this situation.