Can we really intersect circles?

This is discussed in tremendous detail in Greenberg's Euclidean and Non-Euclidean Geometry, which includes the "Circle-Circle Continuity Principle":

If a circle $\gamma$ has one point inside and one point outside another circle $\gamma'$, then the two circles intersect in two points.

Greenberg also discusses the related "Line-Circle Continuity Principle" and "Segment-Circle Continuity Principle". His treatment shows that the first of these principles implies the other two. He also writes (p. 131, 4th edition):

You may wonder why we have called these three statements "principles" instead of "theorems" or "axioms". The latter two would be theorems if we assumed the first one (as we will later show), but we do not wish to call the first one an axiom because we wish to illuminate exactly where it is needed, and then we will ad it as a hypothesis.

Greenberg goes on to note (p. 137) that the circle-circle continuity principle can be proved from Dedekind's axiom:

Suppose that the set $\{l\}$ of all points on a line $l$ is the disjoint union $\Sigma_1 \cup \Sigma_2$ of two nonempty subsets such that no point of either subset is between two points of the other. Then there exists a unique point $O$ on $l$ such that one of the subsets is equal to a ray of $l$l with certex $O$ and the other subset is equal to the complement.

Greenberg notes that the proof of the Circle-Circle principle from the Dedekind axiom is proved by Heath in his commentary on the Elements.

I should add that I think (not quite positive) that Dedekind's axiom is a consequence of (perhaps is equivalent to?) Hilbert's "Axiom of Completeness". So it is not correct to say that Hilbert does not address this.