As a kid, when I first read Harry Potter and learned that he could speak Parseltongue, a language skill one was either born with or otherwise could not acquire, I had automatically thought of people who were born with perfect pitch and related the world of wizardary to the world of music. (They are both magical indeed). There were many sources that said you were either born with it or not, and you could never learn it. I thought the same until I began identifying the notes on a piano with no reference after taking ear training lessons from an amazing teacher. Slowly I began hearing the pitches on brass and string instruments, and eventually on others. I still am not so good in identifying pitches in vocal music as there is a ton of microtones. I also cannot sing a note out of the blue and hit the note I want to sing (yet). But on tuned instruments I can hear the notes right away.
Having trained pitch gave me confidence and ease; I felt like I was accomplishing something I wasn’t naturally supposed to. However now that I think about it, having perfect pitch as a pianist is not necessary to succeed as you don’t tune your own piano (for a string instrument it does matter). But it saves you time in music school when they ask you in an exam what key a piece is in, or when they want you to notate a passage from a symphony. Or when the tuner hasn’t done a good job and you can criticise him.
Even though having perfect pitch does not matter, you must have a great sense of relative pitch to succeed (obviously).
For me, having trained pitch even had a disadvantage at times: I would hear the note names in my head instead of their functions whereas my friends with relative pitch would work harder to figure out the the context of the pitches through thinking about their role in music. A “C” doesn’t mean anything until it is put in context, as it can be the tonic in C the key of C, or a subdominant in G or a dominant in F etc.
When I went to Eastman, I learned for the first time, to sight-sing in scale degrees (which is essentially singing the numbers instead of their names unlike “solfege” where you sing the names of the notes). This changed my perspective completely; I could suddenly hear music mathematically and I was able to spot patterns and analyse music much easier.
Understanding the architecture in music rather than knowing the note names is much more important. So if you don’t have perfect pitch, do not worry!