Hiring a PhD student who left another PhD position

I would contact the students directly for answers before contacting the supervisors. This way, you can get some feedback and further information before deciding if you want to contact the supervisors.

However, I would in any case not take a student from the lab of an acquaintance without the "blessing" of the acquaintance. Otherwise you risk setting up an unnecessarily antagonistic relationship between you and the other advisor.

To get a feel for this situation from the student's point of view, I suggest reading questions such as Changing PhD programs: should I submit a recommendation letter from my old advisor if it's not purely positive?, Incompatibility with the PhD advisor, Switch PhD program: how to contact possible PhD advisors when already enrolled in PhD program?, and many others.

The decision to try to switch programs is a difficult one for the student. They may be worried about harming their relationship with the current advisor if the move does not work out. From this point of view, contacting the advisor, rather than the student, would be like calling a job applicant's current employer.

Although if they asked here they would probably be advised against it, there may be a temptation to just not mention the current PhD program other than to the extent that they have worked as a research assistant.

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The key question you need to ask both the students and yourself is "What would be different about my project compared to your current one?". Of course, there are no guarantees, except that you will not be able to get through a reasonably interesting life without embarrassment.

For example, see How to explain in PhD interview about leaving current PhD due to lack of funding?. If you have good funding, and the student is switching because of lack of funding, there is no reason to expect them to be particularly likely to leave.

The OP for that question wrote "I did not mention my current PhD in CV, and just wrote about my working experience in the last 1.5 year.". I have no idea why students do that, but it seems to be a common urge. Many questions either discuss the advisability of not mentioning incomplete degrees when applying for a PhD program, or are trying to deal afterwards with the consequences of not mentioning them.

Like others have written, there may be uncomplicated reasons that motivate the student to leave (like lack of funding, or a position that was advertised as PhD position but is incompatible with a PhD due to the work-load).

But let's suppose it's a problem with the supervisor:

1. Ask students? Yes, ask, but it's a touchy subject. After all, bad supervisors do exist, and some are surprisingly "good" colleagues. Criticizing a supervisor is usually seen as disloyal (talking among PhD students in a bar excluded). Which is a problem if the loyalty is one-sided and the supervisor is exploitative or abusive. (Then again, there are students who have rather strange ideas what a supervisor should do for them so things aren't always clear-cut.) So I'd ask for their reasons for leaving (Funding? Did things get tough and the grass looks greener here? Do they want a higher-status position?). And if it's an issue with the supervisor, I'd go for a student who states it rationally, sees the interaction as two-sided (even if the other person is a jerk), and -- above all -- maintains a basic level of respect (as a sign of maturity).
2. Ask the supervisor? Like others have written and you suspected, ask the student first. Yes, it's better if things happen in the open, after all, that's the polite thing to do and Academia is small (well, the respective sub-domain is). But there is also a power-imbalance at work here. Here I'd simply ask why they did not include any references (might have been an oversight), and if it's a problem with a specific person, whether there are other people who can be contacted (not being able to work with a supervisor is one thing, not being able to work with the whole department another).
3. Are they quitters? Depends on their reasons for leaving. Actually, I think it's a sign of competence if a person leaves a bad position. The worst situation you can be in is when you are exploited/abused with no chance of establishing yourself as a scientist, yet feeling unable to leave due to escalating commitment/one-sided obligation. So, it is an accomplishment to leave a dysfunctional environment, esp. if you naturally feel a strong commitment to a position.

But yeah, in every hire there is risk, and there are concerns. But I would look at the qualifications and treat the history as one factor (dispassionately, you are not responsible for their situation and they have to fit in your environment). I'd also ask the following questions:

1. What did they learn so far?
2. What do they think they need to finish their PhD successfully (and is that realistic)?
3. Are the habits they have acquired helpful?
4. Can I work with them?
5. What are their plans for the future?