Circles, where do you meet?
Python 3 + numpy, 310 291 bytes

 -19 bytes thanks to ovs! 



v=lambda c,x,y:sin(arctan2(*c[::-1])-x)&gt;sin(y-x)
def f(a,r,w,x,b,s,y,z):
 d=linalg.norm(b-a)
 if abs(r-s)&lt;d&lt;r+s:e=(r*r-s*s+d*d)/2/d;g=a+e*(b-a)/d;t=(sqrt(r*r-e*e)*(b-a)/d*[1,-1])[::-1];return int(v(g+t-a,w,x)and v(g+t-b,y,z))+int(v(g-t-a,w,x)and v(g-t-b,y,z))-(t==0).all()
from numpy import*


Try it online!

This is a horrible answer.

Takes angles in radians.