I'm head of a search committee and will be checking references for finalists for a fixed-term position. Two finalists worked with the same non-profit and gave the names of different people who work there. When I am talking to one person at the non-profit, is it okay to ask what they think of the other person too, or should I ask their opinion of only the candidate who specified them? If it matters, I am in the United States.
In Academia, you do not disclose who has applied for a job unless you have the candidate's permission. Since academics work in teams on long-term projects, sometimes they have to keep their job search a secret. If they do not, they may be excluded from teams.
Ask the candidate for permission before contacting any references the candidate did not provide.
It is common to even ask for permission before contacting references when the candidate provided the references in their application.
Stick with the reference you were given
Are you going to track down ancillary references for all the candidates? You want to be as fair as possible in your faculty hiring. Getting extra references for only a few people is unfair to both the people getting extra references, and people who didn't get a chance to provide an extra reference.
If the extra reference is great, that will likely sway your opinion of the candidate. Other candidates may have plenty of great reference, but you didn't ask them. If the extra was a bad reference, that'll change your mind too - even though you didn't ask anyone else to produce another great reference, or go reference hunting for them.
Job hunting is tough - especially for academics. By asking extra references, you could end up tipping the current organization off about someone's job hunt, which could jeopardize their current job. It also might be illegal (I'm not a lawyer).
There may be law about this, but I would consider it improper unless you ask the candidate first for their OK. Otherwise, you are using a non-official, potentially unfair, informal, "off the record", process to help choose candidates.
You have a defined process. You should stick to it, even if not required by law.
Thinking about the law, defining and publishing one process while using another, might be construed as a kind of fraud, even if it would be difficult to charge.
Unless your search process description (and the job advertisement !) explicitly state that you will solicit opinions from further, secret to the candidate, references, I would stay away from doing so.
It is a thing that makes lawyers panic about the potential of failed candidates suing the university -- at my institution (large state university in the US) I imagine such an action would be considered sufficient procedure violation by the HR department to lead to the search being stopped.