Hello Jack,
From what I've experienced, this filter does not stick from user to user.
I went ahead and tested this on my instance just to make sure and the custom filtered view was not shared amongst multiple users.
To test:
1. Create/Invite a new user (with admin access)
2. Log in this newly created user and change the activity log view for a lead (Lead A)
3. Log in with my original user and went to the activity log view for Lead A
4. The view showed no filtered results
The same goes for when saving a custom view. The custom view is only available on a per user basis.