If your permissions model means you can only access records which are assigned to you, then you can view records which you have reassigned to another User by adding yourself to the relevant record via the Users tab. You need to do this before reassigning the record. 

Scroll across the sub-tabs across the bottom of a record until you reach the Users tab. (NOTE: Not all modules feature the Users tab).

Using the Select User button pops open a window where you can select yourself or other Users to add to this record:

You can then reassign the record(s) in question to another User whilst retaining visibility of those records - they will be visible under the relevant view e.g. "My Projects"

You may also want to read about the options available when automating the assignment of records, please see: https://opencrm.co.uk/help/?kb=1729141&article=How-do-User-Assignment-rules-work