do recruiters just have this info? should I ask?
It's an app /tracking system recruiters use called PRIDEMOD II (PMII). There's a section where recruiter's can verify if you're at board or not, if you're at board there's usually a "check box" thats checked off. When PMII first came up, applicants SSNs were on there... I believe that went away and applicants instead had an "applicant number/ID" instead... so it's a bit easier to share screen shots/etc. to confirm if you so desire.
Also, for PMII, recruiters can only see (1) their own candidates only or if they had DLCPO/OPS access, could see applicants within the NRD/NTAG. So if I was a recruiter at Seattle and you were a candidate in San Antonio, I couldn't verify if you were at board or not.