So, I volunteer for one of the centers that answers calls to the National Suicide Prevention Line (NSPL).
The training that we get involves risk assessing the caller by asking them questions. I won't go into details, but it's very rare that we would call emergency services without the caller's consent, and it would require the caller to tell us that they're in imminent risk of taking their life, and also that they aren't willing to work with us on finding a means of staying safe.
The training that we get puts a lot of emphasis on protecting the dignity and autonomy and privacy of the caller.
There's no one national center, calls to NSPL get routed to the center that's closest to you geographically (I believe that's based on area code). Different centers may have different training and policies, etc.
The vast majority of people who call suicide hotlines do it for this purpose.