In practice no GM (at least that I know) is ever going to question ownership with the first Bestiary. I think that is safe to say.
But it is crazy for anyone but the most casual of players not to buy it. Along with the CRB it forms the backbone of the game. It is also a fun book to own.
On day 1, I purchased the pdfs of the crb and bestiary (I was at a camp and unable to leave to buy the hard copy). Once I could get to a print shop to make a table copy (rather than tablet), I set that up.
I always wanted to play a summoner focused wizard or druid so I knew I needed a copy for the table.
Given how the spells are worded in 2e, any additional bestiaries will be nice as they add creatures to the lists of those available.
Definitely a must have for a summoning PC or a wildshape druid.