i've a question - is your bitcoin legally yours if coinbase holds your private keys?
This is a GREAT question, and really gets to the heart of the matter why Bitcoin is important! Sorry in advance for a philosophical rant...
For all of history, if you owned something and wanted to keep it out of the hands of others who would claim it, you'd need some sort of documentation
outside of that something that proved you owned it. A bill of sale, signature of a notary, in the case of property a deed, or a claim check. Each form of documentation is a contract and up to now the only way to enforce those contracts was via a third party granted a monopoly on force and punishment: eg. Government.
This has given rise to the mindset that if you own something, you must own it "legally." Legally meaning: ownership recognized and enforced by a State. Over time, because we must make obeisance to the State for each thing we own, we've all fallen into the false mentality that ownership of private property is something a State
grants to you instead of simply recognizes and protects. The State itself has capitalized on this shift in perception and all but outright makes claim to everything except what it, in its magnanimity, allows you to keep.
But this mindset is wrong, and we can show this from first principles: Do you own your own body? Almost no one would say no to that; in fact to answer "no" means you support some manner of slavery. If you disagree with slavery, that means you disagree with someone owning rights to the fruits of another person's labour. And if you disagree with that, you agree that the fruits of one's labour is their own
private property.
If you own something, there is no "legally" or "non legally". You either have proof you own it or not. The whole "legally" angle only comes in after and describes who is enforcing that proof if it is ever challenged.
Bitcoin is the first system of money that
bundles that proof with the money itself. No government needs to recognize it, because recognition is implicit in holding the private keys.
So to come back to your original question,
is your bitcoin legally yours if Coinbase holds your private keys? They
are in the sense that Coinbase operates in compliance with the financial laws of a State and if something happened to those Bitcoins you would probably have some legal case to take to the US government. In simple terms, your contract with Coinbase is the third-party proof you need to show anyone that you have access to your coins. But should that contract be challenged, your only recourse is to appeal to the force and coercion of the State to guarantee it.
However, if you transferred your Bitcoins to a private wallet where only you know the private key, the coins are
truly yours in that you own them and you
also have recognized proof via the Blockchain ledger that you own them. No State is required to guarantee this ownership, because the coins and the proof are one and the same! This is precisely why Bitcoin is so revolutionary. It cuts out the middle-man of the State from monetary issues, over which it has become a monstrous monopoly in the last few centuries.
Once again, this statement...
Do you own your own body? The proof that you do is that you're living inside of it.
... is
just as strong a case for ownership as this:
Do you own your Bitcoin? The proof that you do is that you have the private key.
There really is no "legally" in Bitcoin.