To reproduce, do the following tests:
1. Login as User A and navigate to Activities > Scheduling > Tasks > New
2. Fill out the form and set the following:
Assigned to = User B
Reminder Type = Browser
Reminder = 15 Minutes
3. Click Save.
4. Notice that the saved record comes up with Reminder Type/Interval = None
5. Login as User B and check the Calendar.
6. Pull up the Task and notice that the correct Reminder Type/Interval is saved
In a nutshell, the Reminder Type/Interval will only be viewable to the User set under the Assigned To field. This is by design.
Note: This same behaviour applies to Events and Phone Calls.