[RPG] A medusa and a beholder look at each other. What happens

While the general section on beholders in Lords of Madness doesn't give any particulars of the furniture in the beholder's lair, there is an example lair detailed later in the chapter (page 56). I won't reproduce the whole thing here, but the important room for our purposes is the beholder's personal chamber:

This large chamber is quite impressive; the walls and ceiling are covered with intricate carvings depicting beholders eating humanoids, and vast fields of mushrooms with writhing tendrils. On the ceiling above, the image of a single immense beholder watches everything. Two large, uneven stalagmites protrude from the floor along the northeastern wall, and to the south stand several well-made statues of powerful-looking adventurers and fearsome monsters. In the center of the room sit numerous desks, each of which is covered with mounds of papers and books.

This chamber is Sekarvu’s personal lair, and where it spends the majority of its time.

There is no mention of a bed, or indeed any furniture beyond the desks. So either the beholder sleeps on a desk (face it, we all have occasionally), or it continues to levitate while it sleeps.

A possible source of further details on beholders would be the AD&D 2e book I, Tyrant, which was the book in the Monstrous Arcana series dedicated purely to beholders.

If we're going with a strict reading, then if you're seeing the medusa's eyes from beyond its range, then you -- the remote viewer -- do not need to make a saving throw.

When a creature that can see the medusa's eyes starts its turn within 30 feet of the medusa, the medusa can force it to make a DC 14 Constitution saving throw if the medusa isn't incapacitated and can see the creature.

Your familiar will need to make a save, of course, if they are within 30ft of the medusa.

If you are within 30ft of the medusa and use your familiar to see, and you see the medusa's eyes, then it depends on the final clause of that rule: if the medusa isn't incapacitated and can see the creature.

Let's imagine you are a resource-depleted familiar-owner hiding from the medusa in an abandoned building. She is only 25ft away from you, but is unaware of your location and cannot see you due to being in total cover from a stack of crates. Then, you look through your familiar's eyes (a spider) to scout the area, but end up looking straight into her eyes.

In the above scenario, you do not make the saving throw, but your familiar does. It would potentially be a traumatizing event though, being in the mind of a creature as it is being petrified.

Otherwise, if you had failed your Stealth check and the medusa actually can see you, and then you scout via your familiar and end up looking into her eyes through the familiar, then yes you will need to make a saving throw.

Strictly speaking, as long as the four conditions are met, you have to make a saving throw.

1. can see the medusa's eyes
2. starts its turn within 30 feet of the medusa
3. the medusa isn't incapacitated
4. the medusa [...] can see the creature

It doesn't matter where your sight comes from as the first condition is not "that can see the medusa's eyes through its own eyes".

I'm reminded of spells such as Misty Step, which allow you to teleport to any unoccupied space you can see within 30ft of you (even if you're seeing through the eyes of your familiar).

As for the case of controlling another creature, that depends on how you are "controlling" them. If it is through Dominate Monster, for example, then only the creature you have controlled makes the save.

But more likely, you're asking about Magic Jar. If you are possessing the body of another creature in this way (your soul leaves your body and steals the body of another living creature), then from the wording of this particular spell, your soul and the possessed creature's body are considered as one creature. You -- the possessor -- must make the saving throw using the stats of the possessed body. If you fail the save, the body you are possessing becomes petrified. If that body also dies, your soul may leave that dead body and go elsewhere.

As for should your original body be petrified along with the possessed body, that depends on if your true body is within 30ft of the medusa. If it is, then you must also make a saving throw with your actual stats.