If they're discreetly seeking other employment, you have now blown their cover. A couple of things about that:

1. "If they're looking for another job, they can have no guarantee of secrecy." -- Sure, but is it your job to out someone?

2. "I need a second opinion on their references, just in case they're giving me bad ones." -- So.... what exactly is your threat model, here? "Catfished by a candidate" stories are interesting because they're rare.

3. "If I tell other people they're looking, I can reduce their leverage by making things awkward for them at their current job." -- Hey look! I found the sociopath!

Bonus Thing: It's a small industry, and people don't go away. You've now marked yourself as someone who can't be trusted with knowledge about someone's else career choices.