I answered calls for Samaritans for about a year, and answered texts on Shout for about the same amount of time before that. From my own experience, I’d say 1 to 5 lives per day is extremely optimistic, for the following reasons:
The vast majority of callers are not planning to take their lives right at that moment / imminently. People call for all kinds of reasons—e.g. loneliness, bereavement, being in prison and that sucking, trying to stop self harming, etc.
The majority of calls are from repeat callers (and a significant minority are misuse of the service). Only a few are calling for the first time. (Samaritans doesn’t track people, but people often just say they call regularly. Shout did track people. I forget the exact % of conversations that were not first time users but it was definitely most.) And this is obvious right—if someone calls 20 times they generate 20 more calls than someone who calls once.
For people who really are at severe risk, the reduction in the probability of suicide from one call is pretty unclear, but is certainly much less than 100%. Even for someone who eventually works through what they’re dealing with, it will probably have taken many calls, use of other mental health services, reliance on friends etc., probably over months or years.
I’m not aware of any trials of this kind of intervention, but they could be done. E.g. introducing a new hotline service in a country that doesn’t currently have one, but only for a randomly selected half of districts/counties/states, and then comparing the impact on suicide rates over time.
My own unscientific feeling from doing this was that I probably helped a lot of people feel better that day / deal with some kind of crisis, but probably directly prevented very few suicides, if any.
Edit: thinking about the numbers a bit: there are ~6500 suicide deaths in the UK per year. Samaritans has something like 150 people answering phones 24⁄7 (extremely rough). So if every one of those 6500 people calls (absurd) and if the service improved so much they all survived (also absurd) that’s still only 0.04 lives per person day (taking a day as 8 hours). So I think you have to start there and maybe go down a few OOMs due to those absurdly optimistic assumptions.