Let's say I'm attacked with an effect that deals 1d4 persistent damage, and the attack scores a critical hit, doubling all the attacks damage. Does that mean that at the end of each of my turns, I have to role 1d4, multiply the result by 2, and deduct that from my hit points?
That looks correct to me.
I think it is also appropriate to roll 2d4 for the damage instead, though that does have a slightly different distribution of resulting values. But rolling the 1d4 and doubling is the official rule.
You are correct! I play a Bomber and deal out a lot of Persistent Damage. Most of it is in the form of flat values (like 6 pts Persistent Electricity from a Sticky Lesser Bottled Lightning) but Acid Flask Persistent Damage is always rolled.
If I Crit with a Sticky Lesser Acid Flask, at the end of the target's turn it would be (d6+6)x2 Persistent Acid (between 14&24 pts.) Starting next level, a Crit with a Sticky Moderate Acid Flask would be (2d6+7)x2 Persistent Acid (18-38 pts.)
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The Splash Trait suggests doubling the dice when you crit with something that does persistent damage, and I agree. It's just easier to visualize for most people.
For example, if a Swashbuckler hits with their Bleeding Finisher, they deal their normal Finisher damage but also inflict massive Bleed damage to the target. At level 9 this would be 4d6 on a normal hit.
Now if they crit, you could say they have "4d6*2 bleed damage" or just say "8d6 bleed damage".
The Splash Trait suggests doubling the dice when you crit with something that does persistent damage, and I agree. It's just easier to visualize for most people.
For example, if a Swashbuckler hits with their Bleeding Finisher, they deal their normal Finisher damage but also inflict massive Bleed damage to the target. At level 9 this would be 4d6 on a normal hit.
Now if they crit, you could say they have "4d6*2 bleed damage" or just say "8d6 bleed damage".
That's strange, since the rules state you need the GM's permission to roll the dice twice and double the modifiers for crits.
Also, only the higher damage applies if I take two of the same type, right? If I already have 4d4 persistent acid damage and I take 3d6 later, which do I use? Maybe "rolling anew each time" isn't referring to the end of each turn after all?
You do only pick one of the two instances of persistent damage. And you do re-roll the damage each time you take it.
There is clarification on the FAQ page that tells you to just use your best judgement on which damage dice amount is higher in general.
Page 621 (Clarification): How can I tell which multiple persistent damage value is higher when it's not obvious?
Multiple Persistent Damage Conditions notes that you keep only the highest amount of persistent damage. How do you choose between 3 persistent fire damage and 1d6 persistent fire damage (which is usually, but not always higher)? In these cases, the GM should quickly use their best guess to decide which applies. Don’t worry about doing all the math of average damage, just follow your heart to which seems most severe.
Adding to my confusion, the Pathfinder Character Sheet on Roll20 only accepts numbers for this condition and not dice formulas. As if the roll is made once when the condition is given.
Sounds like a bug that you should report to Roll20.
Instead of taking persistent damage immediately, you take it at the end of each of your turns as long as you have the condition, rolling any damage dice anew each time.
It's one reason I'm grateful that Sticky Bombs scale the way they do. Once you hit 10th, there's basically no denying that 1d6+6 Acid (Lesser Sticky Acid Flask with Expanded Splash) is stronger than 2d6 Acid (Standard Moderate Acid Flask.)
At 11th, it's even easier (2d6+7 Moderate Sticky vs 3d6 Greater).