Walpola Rahula uses the example of a child and an old person.
When you are 60 years old, as I am - I know that I was a child of a few months old long ago, and can even say when and where I was born, because my parents told me, and because I have a birth certificate also to help confirm it.
But I have no memories of that at all. And there is almost nothing in common by way of interests, likes, personality etc between me as I am now and that young child.
Actually, not sure you can say that Buddhists "believe in non self" anyway.
If you truly realize non self then you don't need to believe anything, as you have already realized Nirvana. And if you haven't realized it, then what good would a creed do? Because you don't yet truly understand non self. So what is the point in affirming belief in something if you don't understand it? When realizing the truth is something that would liberate you from Samsara?
So Buddha said "come and see". Find out for yourself. It's the only way.
What we have instead of a creed is the possibility of going for refuge to the Buddha's teachings - as one of the three refuges. So that means that you have faith that there is validity in this path for you.
Not that you believe any particular thing. Indeed, anything you believed in that way could not be the truth of non self (or you'd be Buddha already). At most it could be some truth that in some way gives a hint or direction towards the idea of non self. But to think that what you understood is already the truth of non self is to mistake the finger for the Moon in that famous simile.
So it is just that you feel that by following this path that there is some truth there to be found. Which you need eventually to see for yourself..
So anyway - it's not denying personhood either. If it did that then it would indeed make it very puzzling to try to understand how karma could operate.
In some of the sutras Buddha talks about his childhood, and about how he became enlightened. There's no idea at all that it is a problem to do that.
So, that shows, if you accept that the sutras are valid in some way - that you can still talk about "I", and say "I did this" etc when you are enlightened.
And - it is no problem at all saying that I when I was a few months old is the same person as the I that I am now. For instance, if I'd had a major accident way back then, then it's quite possible that I might still be affected by it today even though I have long forgotten the incident itself.
So - it's like that for the connection from one life to another. Just as with the young child, I remember nothing at all about previous lives. But my actions in those lives have consequences today and my actions now have consequences in the future. But not in a deterministic way. What happens to me now depends on the body I have now, the world I'm living in, the people I share it with, natural events that are entirely beyond my control and so on. And obviously what happens now depends on decisions by many other people as well as natural events. So my past karma is just part of that whole picture. It has drawn me to take rebirth in particular body, and attracted me to this particular world and situation in life.
But it doesn't determine what I do or what other people do in this moment. Which is why we all have the possibility of realizing non self, and enlightenment.
If karma was deterministic, and determined your behaviour and other people's behaviour, there would be no possibility at all of escaping from Samsara, because whatever you did would still be part of the cycle of existence and bound by karma.
The details of Karma are very subtle and Buddha warned us against thinking about them overmuch. For two reasons. First it can get us obsessed thinking about who I was in past lives, which is reinforcing that very illusion that I have a self that we hope to see through. And as well that it is so complex and intricate and subtle that you can't hope to unravel how it works and it could drive you mad, basically, to try to sort it all out. Only enlightened beings can in some direct way understand the workings of karma he taught.
Some traditions go into a great deal of detail. The Tibetan traditions particularly with recognized incarnations, and detailed descriptions of the Bardo state. What they say is often quite mind boggling, e.g. several people reborn as one, or one person reborn as several.
And others they hardly talk about it at all, e.g. in the Zen traditions.
And they say things that are not consistent with each other, e.g. Therevadhan Buddhists say that there is no Bardo state - that you just take rebirth immediately after you die.
I don't think we can expect to sort that all out and find a "final right answer". Or indeed to decide for sure from our own experience whether we take rebirth or something else happens when we die.
As a practicing Buddhist, all one needs is an open mind that I don't know what will happen when I die, and that what I do now has effects way beyond the immediate effects I can see for myself, in both time and space.
And then this idea that there is some truth to discover, and that one is following a path that leads to the possibility of realizing that truth, and helping others to realize that truth, in whatever form it might manifest for them, to be able to connect to the ground truth of their situation. Which, I think, doesn't have to be anything we can recognize as non self in the sense of the Buddha's teachings.
But as to what that truth is, I think best to take any ideas one has about that as just hints, and to recognize that if one understood it truly, one would already be enlightened.