Can't select territory with regular user

I am having issues with territories. I’ve made settings for acl roles and acl territories. When I am with user admin I don’t have issue to change the territory in any record. But when I log in with regular user who have rights to see own records and create/edit them I can’t select/choose any territories. Can you help me?

I can confirm the issue with the following clarification: The logic behind “ACL Territories is working fine”. The users see the records they are supposed to see in a module to which territory type is associated. The problem is that in case the user is not an administrator and wants to create a new record in this module or modify the field primary territory in an existing record - he/she can not select a territory from the search list and respectively can not save the record as this field is mandatory. For administrators the search returns the list of territories defined in the system but for regular users this list is always empty.