For webinar marketing, how are the responsibilities split between product marketing and content?

Ooof this is a tough one and honestly it depends on how things were structured at your organization from the beginning. 

IMO, product marketing should influence/contribute to webinars, but not own webinar execution. For thought-leadership webinars, I've seen product marketing, content, and comms all own this in the past. 

If you're lucky to have a large marketing team, I usually divide responsibilities for webinars like this: 

1. Product marketing owns outline and enablement material 
2. Content owns design and copy refinement 
3. Ops or Success owns delivery and lead tracking backend