Thanks. Does that mean I’m supposed to fight over the water? Or should they draw lots over who drinks?

You said there’s a reference in the Talmud—where is it found?

In the Talmud (Bava Metzia 62a) there is a famous discussion about a situation where the water has an owner. I thought that was what you meant. As for joint ownership, there is no discussion there, but the commentators did discuss it. Seemingly, one can infer that the entire dispute is only where the water belongs to one of them. Therefore it would seem that when the water belongs to both of them, even Rabbi Akiva agrees with Ben Petora that they divide it. But in my opinion that is not reasonable, and they should draw lots. To be sure, one can challenge what I’m saying: why didn’t they present the dispute also in a situation of jointly owned water? That can be answered in several ways. Here are two, for example: 1. They wanted to teach that in a situation where the water belongs to one of them, he is the one who gets it and not by lottery. 2. They wanted to teach that even in such a situation, according to Ben Petora both should drink (to show you the full force of Ben Petora’s view).
See my article on separating Siamese twins here on the site.

Understood, thank you very much.
As for Siamese twins, I think there is a difference. With Siamese twins, doing the action that will prolong one of their lives will directly kill the other, whereas in the matter of the water, although he dies because I drank and he didn’t, that is a consequence that isn’t necessary.

I didn’t compare it to the case of Siamese twins. I referred you to the article so you could see a consideration of that kind. There I distinguished between a situation in which the shared organ belongs to both of them and a situation in which it naturally belongs to one of them.
By the way, I’m now thinking that perhaps when Ben Petora said “both should drink,” he didn’t really mean that they should literally divide it, but rather that they should draw lots. His claim is that ownership of the water is irrelevant (as opposed to Rabbi Akiva’s position), because I cannot drink and watch my fellow die just because the water is mine. But he also doesn’t say that both should drink and die; rather, that both of them should have a possibility to drink and be saved.
True, the wording there is “it is better that both should drink and die,” which implies that he means they should literally drink and die. But perhaps one could push the reading and say that it is preferable that both drink and die rather than that he drink alone. But the real alternative is to draw lots. That is a bit forced also because Rabbi Akiva’s opinion is brought after Ben Petora. The Gemara should have brought Rabbi Akiva’s statement first, and then Ben Petora would respond and say that if you say the owner of the water should drink, then it is better that both should drink and die. And therefore they should draw lots.
Anyway, just a thought that occurred to me now.

But seemingly, if drawing lots is good, then isn’t ownership of the water no less a good way of winning the lottery?

Absolutely not. A lottery gives both of them an equal chance, and therefore both of them would agree to it. Such a lottery is an implementation of “both should drink.” In another formulation one could say that now, after we have divided the water according to Ben Petora’s view, we agree to draw lots, and whoever wins gets the entire flask. Instead of dividing the water, we divide the chance to win it. Each of the two would agree to that, since instead of getting only temporary life, he gets a 50% chance at full life. Exactly as I described in the article about Siamese twins.
The distinction between ownership, which is a state of affairs that already exists when the problem arises, and a lottery that is carried out now, is also discussed in column 538.

And wouldn’t both of them, in principle, already agree from birth, behind a veil of ignorance, to a rule that whoever owns a flask of water gets to drink it?

Not likely. It’s a hypothetical case. It’s no better than a mere asmachta.

Another formulation: by the same token, both of them would also agree behind a veil of ignorance that the flask should be given to the one to whom it does not belong. They would also both agree that a lottery should be held in real time. So why are you choosing specifically the agreement that the water be given to the owner of the flask? Because the power is in his hands. We’ve gone back from lottery to power. See my above-mentioned columns.

Since Ben Petora also gave a reason for his view, you can’t say that he would in principle also agree to a lottery, because his reason is that one should not see his brother’s death, and I have no permission to agree to a lottery that will let my fellow live and kill me—perhaps because then my fellow will see my death, and that contradicts Ben Petora’s principle, since he himself relies on verses according to which I am explicitly forbidden to leave my brother to die.

And one could say that along these lines, it is possible to argue about what Rabbi Akiva’s opinion would be in a case of partnership. As we see, Rabbi Akiva did not dispute Ben Petora’s moral premise that it is preferable not to watch my brother die. Rather, he disputed that this moral argument applies where my right to life (my ownership) takes precedence over his, and then I have no obligation to forfeit it for the sake of his right to life (the exposition of “that your brother may live with you”—that I need to see to it that he live with me, but not to lose my life for that). But where my right to life is not preferable to my fellow’s, the moral argument remains valid, that it is preferable not to watch my fellow die. Ben Petora is not interested in the fact that it is preferable for one to die rather than two; he is dealing with the morality of not having one person live and watch his brother die. Of course this is only a thought—I’d be happy to hear your opinion.

If the problem is finding some preference for choosing ownership, that seems to me a pretty small problem. Giving preference to the person who owns the flask gives each person, in advance, the possibility of preparing for the situation and relying on himself (as most reasonable people want). In a situation of drawing lots or giving the flask to the non-owner, he would basically have to check from the outset that everyone in his group has water and that they aren’t wasting it. That’s a much heavier burden. A more sensible distribution of the burden is that each person takes care of himself, and that way in the end everyone benefits the most. Therefore, behind a veil of ignorance, it seems likely everyone would agree דווקא to that. Therefore it seems to me that if Ben Petora is satisfied with a lottery, then it is unnecessary.

I don’t understand the question. I already explained it. Ben Petora answers Rabbi Akiva that one should not watch the other die, meaning that the owner of the water should not drink and watch the second one die. But a lottery is a different matter, since each of them is given a chance to live.
I also didn’t understand your second remark. You’re repeating what I said. After all, that is exactly what I wrote as the straightforward reading of the passage: that from the fact that they disputed in a case where there is ownership, it implies that where there is no ownership even Rabbi Akiva agrees with Ben Petora. Afterward I rejected that.
I’m not sure I understood your third remark either. Is it referring to the implicit lottery proposed here? According to your proposal, that is not a lottery but a different consideration. If you were proposing a lottery or agreement on ownership, I do not know why everyone would prefer to agree to that. So in practice it seems that what you are really doing here is applying considerations of power in disguise.