Yes, programs can see even those who didn't match.
I just looked at the printouts from this year. According to the date stamp on the lower right-hand corner, it was printed March 15 (Match Day). We have printouts for all candidates we ranked (both those who matched elsewhere and those who did not match at all) except for those who matched at our program.
Each page has the following data fields:
Applicant
Name: XXXXX YYYYYY (NRMP number)
School: ZZZZZZZZZZZZ (Med school number)
History of Matches
NRMP Main Residency Match
Status Matched Program Matched Institution Year
(Matched or Unmatched) - Lists Intern year first, advanced second
Waiver
(No waiver found or waiver found)
Violation
(No violation found or violation found)
If a candidate did not match, it says, "The applicant did not match in any previous NRMP match. The applicant may be offered a position in a future match."
Hope this helps!