Take into account that reincarnation in a lot of Eastern countries is widely accepted, so children in those environments grow up with that in mind. I don't think whether you talk to your children about reincarnation or not will influence their thoughts, really. I think a lot is to do with the parents, and how receptive they are. Many kids may talk about past lives/exhibit strange behaviours and if parents are unaware of reincarnation they will totally miss the cues.
I've talked to my oldest child about reincarnation, because he had heard me talking to his father about a person called John, and he was curious who John was. That was fairly recently though, and he's now seven years old. He understood the concept of reincarnation without me having to explain it much, if that makes sense. My son when he was around two told me spontaneously that when he had lived before, he had been a 'funny man', and lived in a big house. At that point I had never talked to him about reincarnation. He's also had some pretty bad nightmares about being in a car crash since then, so do wonder if that is all related. However he seems to have forgotten about the 'funny man' and also the nightmares for now. By 'funny man' I assume he means he was a comedian, and the big house makes me think he made a living from it - but that's my assumptions there which I'm not going to impress on him.
My middle daughter has exhibited spontaneous pl memories. She is a very lively child, not afraid by most things, but she can not stand fireworks. When she was around 20 mos. old we had fireworks in the back yard. She became hysterical, ran in the house, and grabbed the nearest thing to protect herself, which happened to be my husbands bicycle helmet. When she was as little as 12 mos. she took to a sieve I had which was very much like the shape of the old Tommy helmets, and would place it on her head. She had some play eggs and liked to rub them against her jumper when she was as little as 12 mos. old, something that someone bought up in a rural environment might do. Me and my husband had never done that before seen as our eggs are store bought. She would methodically take each egg, clean it and place it back in the plastic box. I've also caught her on more than one occasion spit shining her Peppa Pig boots -- again never done that in front of her, and have no idea where she got the idea of it from. Again never mentioned reincarnation to her then, and she too hasn't opened up beyond that. I have a wait and see approach -- I guess one day she might want to open about it, either in her childhood or later. She still becomes very distressed by fireworks, in particular the loud rocket types that 'woosh' then 'bang', she cries and clings to me if she hears one go off. She's still too little for me to explain fully to her reincarnation, in my opinion.
I do wonder if I would have missed those cues with my daughter if I had not known about reincarnation myself first hand.