How to triangulate 2 sources on a 2d plane formed by 4 sensors?
Conceptual partial answer:

You can use 3 mics to find position in 3D space, except you can't tell if it's above the plane or below the plane (unless you already know there's nothing above the plane, of course).

Since your 4 microphones are in the same plane, the 4th microphone is redundant and doesn't help you know whether the source is above or below the plane.  (Though it improves SNR/accuracy of estimate, as would any number of additional mics in the same plane).

There is a delay due to speed of sound from the source ($S$) to each microphone.  The delay difference between each pair of microphones gives you a hyperboloid (green cross-section) that the source must be on, in order for the waves to be delayed by that relative amount (due to extra distance $a$).  Each pair of mics defines a hyperboloid, so by combining 3 pairs of mics, you can find the intersection of the 3 hyperboloids, which will define a unique set of 2 points.  (Symmetrical above and below the plane.)

(4 mics gives you 6 pairs of microphones, so by averaging the intersection point of all 6 hyperboloids you can improve the accuracy of the estimate, but you still don't know if it's above or below the plane.   With a tetrahedron of mics you would only have one intersection point.  You could probably weight the average, too, since some source positions will be more accurately measured by some pairs of mics, etc.)

Also, the sound pressure drops off as 1/r with distance.  The attenuation (amplitude) ratio between each pair of microphones gives you a sphere (yellow cross-section) that the source must be on.  By combining this information from every pair of microphones you can find average intersection of all 6 spheres and improve accuracy from that, too.



The spheres and hyperboloids are usually not tangent at the intersection point, so they each improve accuracy in a particular direction:



To find the relative delay, you can use cross-correlation of the signal from the 2 mics.  There will be a peak at the "lag" at which the two signals line up with each other.



To find the projection of the source location on the plane itself, just throw away the z coordinate.

See https://en.wikipedia.org/wiki/Multilateration#Solution_algorithms for more details on actual implementation.