EMB Legacy pilot reported a CFTT event after receiving a low altitude alert from ATC during a localizer approach. Flight crew continued to landing after receiving a visual approach clearance.

Narrative

We were cleared for the LOC DME instrument approach into ZZZ and were instructed to report the field in sight. At the time we were cleared for approach we were IMC in a scattered or broken layer of clouds. Shortly after ZZZZZ intersection at 13;000 ft. (our approach clearance altitude) the Pilot Flying (PF) says he has the runway in sight. I look up and do not see the runway in sight; all I see are clouds. I look back down at the Primary Flight Display (PFD) to orient my brain to where the airport is to try and find it. By the time I look back up; the PF had disconnected the autopilot and rolled into a right turn and descended in excess of 2;000 FPM off course. Roughly 10 seconds later; tower urgently tells us a low altitude alert and to check our altitude immediately. At this point during the descending turn; I do make visual contact with the runway and call field in sight. Tower clears us for the visual at this point and shortly thereafter instructs us to remain west of the runway center line. We comply. The PF continues to descend lower into the valley below surrounding ridge lines. A G550 was opposing us departing the airport as a note of other traffic in the area. We continue the approach and got to the 500 ft. stabilized call out and landed. Tower advised a possible pilot deviation and issued a phone number to us. ATC issued a possible pilot deviation and a phone number after we landed.Cause: The PF/Captain (CA) executed this maneuver without communicating to the Pilot Monitoring (PM) what he was doing. We accepted and the PF/CA called the phone number. I was not present for this conversation.Suggestions: This event needs to be reviewed and discussed with the crew.