Definitely not. As PP's have said - if I thought I needed to hire a PI to check someone out, I wouldn't continue the relationship.
I think it's more about taking things a bit slower - getting to know someone well before moving in with them, letting them be alone with your kids, having kids with them etc. I'm not naive, I know some horrible people who have led double lives....some made more of a mess because they decided to have a child together 2 months after they started dating! You do not know a person that early, and the lust hormones are still firing on all cylinders.
And "checking into someone" can sometimes be a bit like snooping their phone - you find a one sided story and then can't ask them about it, because you're in the wrong for snooping.
If I ever found out someone hired a PI to follow me, or "researched" me before they decided to date me, I'd throw them out on their ear quicker than they could say boo. I would find that akin to cheating on me. Dishonesty is not something I will tolerate. It would make me think the type of person they are is someone who will not communicate honestly with me and will turn to snooping as a first way of getting information.
Also - my past is very different to my present and my future. No, I haven't murdered anyone and I'm not a criminal - but there are things that if my now DH had found out through "snooping" instead of me telling him the story, he may not have wanted to give me a chance either.
For the people who say they would do it - how would you feel about it being done to you? (legitimate question, not having a go).