[RPG] How far can a ship “see”

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.

So I have not used Roll20 in a couple of years (or play D&D over those years unfortunately), but here is what I used to do for my players in situations where they were in the dark without a light (or otherwise had very low visibility). I would turn on Dynamic Lighting but turn off Global Illumination. Then for each of their tokens I would go to the advanced setting, make them emit light out to 5 ft and unselect All Players See Light. That way each player could see what was right next to them but would not see the other players if they were far apart. I also asked my players to not move too quickly (especially outside of combat when they are not taking turns) and they were pretty good at stopping or asking questions when they butted up against something or someone. Also you might try to incentives them asking questions in ways that do not reveal information to other players. "What am I seeing next to me?" instead of "IS THIS HIDEOUS CREATURE HOSTILE!?". In 5E they have ways to reward players for role playing well which could be adapted for this.

The disadvantage of this approach in your case is that if you only want small sections of the map not visible you will have to create a grid of light sources across the map and then turn off the ones where the Obscuring Mist should be.

That is what I have done in the past. Looking at the current offerings I think Advanced Fog of War is what you want. First of all, you can set the sight for each token (under the token's advanced settings):

tokens that emit light, or have a set view distance, clear fog dynamically as they're moved across the Page.

And they can only see what is in view for their token:

The fog cells that a token has revealed is saved for each token on a Page, so each player potentially can have a unique view of the same Page they are all currently exploring.

My suggestion would be to turn on advanced fog of war, set each token's sight to 5ft or however far you think they should be able to discern, use the Reveal Areas tool to reveal the whole map, and then use Hide Areas when something like Obscuring Mist is cast:

If you choose to reset the fog or manually hide areas, the areas within the Token's set sight or vision range can not be hidden from view.

Lastly, if the players need more than what they can see from the map when they uncover an area, I would "whisper" to that player. To help me be able to whisper fast enough I type up (in a text editor or word processor) whispers for things I know I want to share at some point. Then I can cut and paste them to the appropriate players. Of course the players are always going to ask about something that I have not prepared before hand, but hopefully these are not too frequent and all the players are patient while I type the answers to these questions.