If from a purely logistical sense of booking meeting rooms and sending out Outlook invites or a poster on a staff board somewhere, then anyone can do it. It is likely one of the “active” members who runs/presents the session.
If in terms of getting the whole thing started, I briefly touched on this during one of the answers I gave.
First, ask WHY have a Community of Practice? Regardless what they are practitioners of, if they don’t know why they want to come together as a group, it will struggle to succeed. Once you know why you exist, you can think who would allow that to happen. For a Test Community of Practice, you might only want those who perform testing as their primary function, but as the community matures it could look to bring others in who might have some input, or benefit in learning from the community.
Then, think about what would be a BAD example of a Community of Practice as a community. This will allow you to agree what will be done by the community to prevent that from happening. It reinforces it is about the community, and not whoever leads it (whether it is an individual or a small leadership group).
For the sessions themselves, it can depend on what it is. A group discussion will be setup differently from a presentation, which will be different to a workshop.
Hopefully that answers your question, but feel free to reply and seek clarification if not