Can a Responder escalate an incident assigned to someone else?

No. A Responder can escalate only incidents that are assigned to them.

Are escalations recorded?

Yes. Level changes appear in the incident timeline, and a note you add is saved in the incident notes.

No. Oppex contacts the assigned user by email and phone call only.

This article lists who can use each option in the Escalate To dialog, the limits that apply, and what each message means.

Action	Who can do it
Change Level	Owner and Global Admin: any incident. Admin and Workspace Admin: incidents on services they are an admin of. Responder: incidents assigned to them.
Assign User	Owner, Global Admin, Workspace Admin and Admin: any incident. Responder: incidents assigned to them.

When you can't escalate an incident, Escalate To doesn't appear in its ⋮ menu.

Rule	Detail
Incident status	Any status except Resolved. Resolved incidents can't be escalated.
Assign to User tab	Appears only when the option is enabled for your workspace. Without it, the dialog shows only the escalation levels.
Levels you can choose	Any level except the current one, higher or lower
Users you can choose	Any user in your workspace except yourself. The list shows up to 20 users; search needs at least two characters.
Status after Change Level	Acknowledged and Snoozed incidents return to Triggered
Contact after Change Level	The on-call person at the new level, immediately. Escalation then continues under the same policy.
Contact after Assign User	The assigned user, once, by email and phone. The escalation policy stops for this incident.
Service after Assign User	Shows the assigned user's name. With Follow this incident, the original service still receives status updates.
Notes	Optional, Change Level only. Saved in the incident notes.
Undo	Not available. Escalate the incident again to hand it on.

Message	Meaning	What to do
"<name> will be contacted immediately."	Shown after you pick a level. It names the person on call at that level.	Click Change Level to confirm.
"Level changed to Level N"	The level change succeeded.	Nothing.
"Incident assigned to <name>"	The assignment succeeded.	Nothing.
"No escalation levels found for this incident."	The incident's service has no escalation policy for the incident's severity.	Assign the incident to a user, or ask an admin to set up the policy.
"No users found"	No user in your workspace matches your search.	Check the spelling, or search by email.