I hear people talking about anonymity with various currencies, eg Monero, or DASH, but as most people seem to use exchanges, there really isn't any anonymity, because the exchange presumably knows each person's identity and therefore has a record of all transactions, whether they be for bitcoin, or any other "anonymous" currency. Is this right? It would seem that anonymity is a function of the exchange and not the currency (granted there are differences in currencies that impact anonymity as well).

If the above is true, are there any legitimate (I know this is subjective) exchanges that do not keep identities of their customers? I have heard of local meetups, but that does not seem very convenient and there are trust issues.