Is there a way that multiple people can see when a project is assigned/updated/completed within Todoist?
Unfortunately, it's not possible to receive such notification. You can of course post something in the task's comments and assign multiple people to it (or everyone) and they will be notified about the comment, but if someone assigns a task, only the person it's been assigned to can receive a notification.
Also, when the task is completed, only the person that assigned it will be notified. We may consider adding more notification options in the future, though.
Consider adding a "teams" feature. Where people can be grouped into teams and thus would get notifications as a team and individually.