Product managers go to a lot of meetings. The number of hats a product manager wears is one cause that adds to the meeting load.
This is important: We often accept unthinkingly the number of meeting and the amount of time spent in meetings as proxy measurements for effective collaboration. This situation arises when we don't know how to measure outcomes. Individually, we tolerate being involved in so many meetings - even when we don't know why we're in them - because on some level we feel them to indicate importance.
This has to stop.
I don't know how I would survive without blocking off 2-3 hours daily of focus time. These times aren't for administrative tidbits - that happens in the little bits of time between meetings. These 2-3 hour blocks are sacrosanct and I decline meetings scheduled for me during them because I need them to accomplish the reading and thinking involved in strategy-level work and solving hard problems. These problems require what Cal Newport calls deep work, and deep work requires concentration, and concentration requires time.
At first, coworkers didn't get it, but eventually they started respecting them and even doing likewise. My blocks tend, right now, to be mornings before my West Coast colleagues come online.
Additionally, once I'm ramped up and onboarded, settled into a role, I question every meeting I am pulled into if I do not understand my role in it. I ask the organizer what, specifically they expect me to contribute. What takeaway will I have that I cannot get from the meeting notes? If they do not have a clear, specific answer, I make a deal with them: I will use the time block for getting my own work done, but I'll keep myself available in case they need to pull me in.
See what I did there? I allow their meeting, which I decline to attend, stay on my calendar and shield me from other meetings.
You're welcome, Internet.
When people question either action (blocking or declining meetings), I just tell them the reason. If they plead or cajole, I smile, act sympathetic, and repeat the reason.
The great thing about this approach is that you never have to announce it or get permission. You can just start doing it. Your individual coworkers will just start understanding. You can decide to exempt certain coworkers (say, your boss) if that seems more prudent overall. Additionally, it will let you work out your negotiation skills when people start whining and you need to make them feel good while still defending your time. I recommend Never Split The Difference by Chris Voss to improve that skill set.
This is my third company using this basic approach. It works and I expect it to work here. Because I manage in these ways to keep almost half of any given day meeting-free, I produce quickly enough that I think people don't question my how. This production is part of how I produce good results.