Test file:

~~~
class C {
    private C(int x) {}
    private C(String x) {}
}

class Test {
    C c = new C(0);
}
~~~

% -> `javahome 15`/bin/javac C.java
C.java:7: error: incompatible types: int cannot be converted to String
    C c = new C(0);
                ^
Note: Some messages have been simplified; recompile with -Xdiags:verbose to get full output
1 error

Obviously the most relevant problem is that C(int) is private, not that C(String) has an incompatible type.

Reordering the declarations prompts the expected output:

~~~
class C {
    private C(String x) {}
    private C(int x) {}
}

class Test {
    C c = new C(0);
}
~~~

% -> `javahome 15`/bin/javac C.java
C.java:7: error: C(int) has private access in C
    C c = new C(0);
          ^
1 error