Can ssh jump work with keys on different machines - local and bastion instances

I have a bastion machine for which I use the key on my local machine:

local$ ssh -i ~/.ssh/my_private_key user@bastion

Then there's a private key on bastion machine that can be used to connect to private instance

bastion$ ssh -i ~/.ssh/bastion_private_key user@private

However, when I try to do a jump, it doesn't work:

local$ ssh -i ~/.ssh/my_private_key -J user@bastion

Am I misunderstanding the jump logic?

All tutorials say that all the keys should be on the local machine, but can't it work if the key to access from bastion to private are placed on bastion? Isn't it more logical?