I am of the opinion that whoever calls a meeting should be responsible for ensuring that Meeting Minutes are circulated to all attendees.
It is a good habit to get into, and can aid in reducing issues being relitigated and having a record of when decisions were made. It does not need to be an onerous task and only the key points need to be captured.
Meeting minutes should be circulated within 24 hours of a meeting occurring, whilst the information is still fresh in people's minds; I often strive to get them out within an hour or two of meeting completion.
I use a standard (yet simple) template for my meeting minutes which aids in me being able to quickly type them up. I tend to in fact usually type up the vast majority of the meeting minutes during the meeting and then tidy it up and distribute afterwards.
If it is a regular meeting I usually have the first agenda item being agreement of the previous minutes, and sometimes will include the full list of open issues, risks and decisions, although have found that these tend to be better managed when stored externally.
My Simple Template
Meeting Minutes Template