This is a very common question with any kind of copy creature. The most important thing to note is that the static ability that lets you choose to copy something is NOT a triggered ability (if it was, pretty much every Clone-type creature would die for being 0/0 before you could choose to copy something!). It’s a replacement effect that modifies how it enters the battlefield.
So what does that mean for you? If you let a Phantasmal Image resolve, the choice to copy something is made immediately as it resolves and enters the battlefield. This choice doesn’t use the stack and you can’t respond to it. If they copy your Elesh Norn, for example, they will both go the graveyard before any player gets priority again. This also means if you copy something with an enters-the-battlefield triggered ability, it gets to trigger exactly as if that original creature had just entered.
Also note that you don’t have the right to know what will be copied before the Image resolves. In fact, if you ask about a choice not normally made until resolution, it means you just passed priority and are allowing the spell to resolve, thereby giving up your chance to respond to the spell.
But what if my opponent just casts Phantasmal Image while saying “Copy my Sun Titan“? Since they already made the choice, is it too late to respond? Nope! There is a default tournament shortcut that says “If a player declares a choice for a spell not normally made until resolution, that player is held to that choice unless an opponent responds.” So, if you say “Sure, that’s fine” then the player can’t suddenly pull a switch and say “ha ha! I’m actually copying your Elesh Norn!” On the other hand, if you interrupt that shortcut by casting Doom Blade on the prematurely-named Sun Titan, then the player is no longer held to that choice (especially considering that choice isn’t even possible anymore, though the shortcut would be broken if you responded by doing anything at all), and is now free to copy any other legal choice (if there are any).
Today’s Rules/Tournament Tip written by
Josh Stansfield, Level 2 judge from Orange, CA