On Asset notifications, what level of permission should the person being notified have (e.g. Community Manager, etc.)?

No special permission is needed.  This may be defined by the client, as long as the user exists in Forum. 

Example:  You may choose to notify an executive assistant that monitors the Board Room to be notified when someone books it. Or, you can choose to a user when catering or equipment is added to a reservation. The user can be a Registered User with no additional rights needed. However, if they will be approving requests because these assets require it, make sure that they belong to the user group that can.