I need to get some more experience with writing grant proposals, and I know that professors are often invited to review them. How can I get involved in it while still being just a graduate student?
Reviewing grant proposals as a student is tricky, at least in some fields. The NSF is highly unlikely to ask a grad student to review proposals (it might theoretically be possible for a brilliant student who is almost done with their Ph.D., but I've never heard of it happening). Furthermore, faculty are not allowed to show proposals they have been asked to review to their students (it may sometimes happen, but it's breaking explicit rules regarding confidentiality). Overall, in pure math grad students basically never review grant proposals. I can't speak for other fields, but I'm skeptical that grad students ever play a major role in reviewing proposals.
Instead, I'd recommend asking your advisor to see the other side of the process. They could share their own proposals, and perhaps even reviews of those proposals or drafts of upcoming proposals. They could also ask collaborators whether they had any proposals they would be willing to share. This isn't quite the same as reviewing proposals yourself, but it could still give you valuable experience with how grant applications work.
Receiving an invitation to help out at a NSF or NIH study session is highly unlikely and logistically difficult. However, I would try to get involved with study sessions for an University driven call for proposals. Alternatively, you could get involved with a course that does a mock grant proposal as part of its coursework. I just submitted a mock proposal recently and I'm sure that the TAs will be forced to look over my work and get practice in the process.
You can't get experience serving on a grant proposal review panel as a grad student. Sorry; that's just how it works.
However, there are other ways to get experience with the grant proposal process. The number-one way: talk to your advisor/PI and ask them. In particular, ask them if you can be involved with the next grant proposal they write. Ask them how you can help. Maybe you can read a draft and offer comments. Maybe you can brainstorm with them. Maybe they can outline a piece of it and you can try writing a draft of a section.
Also, you can ask to see copies of past proposals they've submitted (both funded and unfunded). After you read the proposals, you can ask your advisor for his/her own assessment of the proposal, and even ask to see the reviews of those submissions from review panel, compare to your own assessment, and use this feedback to improve your knowledge.
Quals proposals are another great form of practice at this sort of thing; they require some similar skills. Spend time on your quals proposal and try to make it outstanding. Read other great quals proposals. Offer to give feedback to your fellow students on their quals proposals. Learn as much as you can from that process, as some of those lessons will carry over to help you write better grant proposals. Similarly, getting good at writing a research statement (for a job application or a fellowship application) is a useful skill that has some overlap at formulating and writing grant proposals. Of course, these are not the same as proposal-writing, but the experience will serve you well.