Multiple assignees does not work in real life, so JIRA simply does not allow it. There's nothing to stop many people logging work on an issue though, not unless you restrict it to assignee only (most projects don't do that, they allow for all developers in the project to help out at least)
You can easily set up associated users though. So that you have one single person who clearly owns it, but several who can also be asked about it, expected to work on it and so-on.
The most simple approach is a multi-select-user field for associated users, but you could also choose a group by using a group picker (assuming your groups are always going to match). There are other variations on those, but start with them to see how they might work out.
An uglier option which allows for errors is to have a dummy user that represents a group of users.
I suspect your permissions are not set up the way you need.
Have a look at the project permission scheme. My best guess is that it's going to say "Can log work: assignee". So that only the assignee can log work. Nothing wrong with that, but it's not working for you.
For your case, I would change it to "Can log work: assignee, and custom-field: your-multiuser-field". That should allow the assignee or anyone named in the field to log work
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