Riemann surface for square root function

I think the problem lies in carefully building up the Riemann surface for a multi-valued function. What I try to roughly explain now, is from Elias Wegert's "Visual Complex Functions", where it is explained much more precisely. I am not an expert, so its just a try to give you some intuitive idea.

First of all, you need to think of the multi-valued function as a large set of function elements, which are pairs $(f,D)$ of (single-valued) analytic functions $f$ and disks $D$ in $\mathbb{C}$. You can use analytic continuation to get "chains" of those function elements building up your function. But since the function is multi-valued, you will get "contradicting" function elements at some point, which is why you lift the function to a number of copies of $\mathbb{C}$. You need as many copies as you have equivalence classes of "non-contradicting" function elements, and you can glue them together to get a nice structure.

Returning to your question, you cannot really speak of the function's values on the intersecting line, because the function on the Riemann surface is built up from function elements. Recall that those are disks, so either you have a disk in the red colored part or a disk in the blue colored part. Depending on that, you know the function's values (red or blue) on the line.