Being a freelance product manager is akin to being a freelance developer.
A product manager works as a contractor assisting companies and individual clients with accomplishing their goals. Whatever these goals may be.
In many cases freelance product managers work on shorter term engagements.
Being that a freelance product manager is not a full-time employee, a large amount of their time will be spent speaking with prospective clients and going back and forth regarding the details of the engagement.
A large amount of their time will be spent trying to acquire their next project.
The nature of their product management work will vary from company to company. One company may require assistance building their MVP whereas another company may need help with performing market and customer research.
Working as a freelance product manager can be rewarding however to do this effectively it helps to have a wide professional network (for referrals) along with experience in sales and marketing (to add prospective clients to the funnel).
Freelance product management is not as common as freelance development. There is more demand in the software industry for freelance developers and designers than there are for freelance product managers.