Would another contracting job (DoD) know if someone accepted a position with another contracting company (DHS) ?

I know a guy who currently holds a DoD Secret clearance. Another contracting company with DHS offered this person a job, which requires:
“active CBP Public Trust or transferable BI (Top Secret, USCIS, ICE, or CBP)”

So, would my friend’s current job be notified/aware that the DoD clearance is being transferred to the DHS job?

Yes. The employer can see who is sponsoring a clearance. I had a Subject who did not notify his employer and they found out from looking at the clearance in the system and the Subject was asked to resign due to a conflict of interest.