How do singers know if they hit the right notes when singing acapella? You can't check against the piano as you are singing acapella.
Once the starting note is established (pitch pipe, or just maybe a member with perfect pitch) they use their sense of relative pitch. Players of all instruments that CAN be played out of tune (i.e. most anything you blow, or where there's a string and a non-fretted fingerboard) need this. If you don't KNOW what the note's going to sound like, how do you know you're playing in tune or even hitting the right note?
Singers need it most. Trombonists and violin/viola/cello etc. players need it a lot. Clarinet, flute, sax need it a bit - the right fingering will get you in the ballpark, but you still need to get it in tune (listen to a kids' band to prove the point!) Pianists and guitarists don't need it at all - finger the note, you get the note. (Though it enables them to confirm they DID hit the right one!)
A capella groups often slip flat during a song. They rarely slip sharp.
More often than not, someone will play a reference note — on a harmonica, pitch pipe or the like — especially if it's a group, like a barbershop quartet. Once that's established, everyone keeps (sort of) in tune with each other. However, because we tend to sing not using 12edo, there's a slight tendency to end slightly lower than it started. One singer by themselves is usually so used to singing the songs, that they sort of internally pitch.