I think it is the end user who asks the support role to agree to share what is on the screen.
Of course, I understand that the support person would find it easier if she could still share the screen.
But I would not be comfortable if someone out of the blue was asking for screen sharing with me.
You should always try to have a dialogue with your client and for me, the "screen sharing" should be the last option; like asking someone to come to your house and help you.
If you can not help the customer via chat; the service representative could then suggest to the customer that he / she shares the screen.
To help someone, the customer must feel good and have a final result.
Some people may be offended if you "go to the action" and ask for screen sharing without asking any questions.
So keep that in mind!