Is this party very far away, as in overseas?
This used to happen a lot back in the days when a lot of communications went via satellite or microwave, but since most have transitioned to fiber optics this is rare. The cause is the radio wave going up to the satellite was partially reflecting off the E layer of the ionosphere (about 75 miles above earth) and the primary wave reflected off a geosynchronous satellite (about 22,000 miles above earth). The reflection off the E layer arrives faster than the reflection off the satellite because of the extra time it takes to travel 22,000 miles up and another 22,000 miles down.
This introduces a delay of about 0.25 seconds, enough for the human ear to detect an echo. Since the atmospheric ionization on your side of the call might be different than on their side, only one party hears the echo. This is especially true if it is morning on your side and the sun is shining brightly and charging up the ionosphere, while the receiver is in the place where the sun is fading later in the day. As the sun fades the ions dissipate and the layers of the ionosphere recombine.
That does not help your problem, but that is a possible explanation.