Firstly, very few people on the service I volunteered for had authorities called on them- roughly one in every thousand calls.
Usually, if I mentioned that emergency services would need to be called, this was actually a motivator for the person to work with me and come up with a strategy to keep them safe for the night.
For those few who did need to have authorities, they fell into one of two categories. Firstly, while rare, there was one person where we had to call the police because they knew the system well and knew that if they told me certain things, it would be required that the police would be called. I can't reveal anything about the call, but they told me that they didn't want to be in the situation they were in, and that they'd been in and out of hospitals before. They understood that getting emergency services involved would get them out of their current situation, which happened to be high school.
More commonly are people where they genuinely needed help. My feeling about them was that if someone uses a suicide hotline, they are asking to be convinced not to end their life. They're literally asking for help.
What upset me more are the times when I strongly felt someone needed to have emergency services called, but they didn't check the boxes, so we couldn't.