How is the weight of a planet determined?

Answer

In the same way as on Earth! We can determine the gravitational pull exerted by a planet from the movements of the satellites (moons) of that planet. The more mass a planet has, the faster the satellites have to move in a given orbit to describe a stable orbit. If we know the distance from the planet, we also immediately know how big the orbits are that satellites describe around the planet; since we can also determine the orbital period, we also know how fast the satellite is moving in its orbit; from that speed we can then, using the laws of Kepler or Newton, determine the mass of the planet.

The method therefore only works if a planet does indeed have one or more satellites. In our solar system, this is the case for all except Mercury and Venus. Historically, the masses of those planets have been estimated from the magnitude of the disturbances they exert on the orbits of the other planets. But those effects are small, and the masses so determined were not particularly accurate, especially in the case of Mercury. But now those planets have (or had) satellites, namely the spacecraft that we have sent there. Accurate masses for Mercury and Venus have therefore been determined from the orbits of these satellites, using exactly the same method as for the others.