Realistically, if he made these trades OTC versus his prime broker, only the prime broker (ie, an investment bank) would know. That's how many of the biggest trades are done. The bank may lay off the short vol slowly, or cover it with several strikes and maturities using automated tools. The only time a bank would hedge aggressively is when they expect the customer to come back and do another piece of the same trade. Presumably if they thought the trade were extremely toxic, they would have passed on it.
If the contracts were listed, option traders would look at big trades that hit the tape without contingent stock printing simultaneously, and call the brokers that crossed those trades to ask which bank sent the order to the floor. Then they'd compare that against banks which are believed to trade with Burry, and filter to get a guess at how much he traded. This may be tough with TSLA because there is so much activity, but in smaller names where only a couple of big prints go up daily, it's pretty easy to figure out who trades what. The banks that trade options against big players get quoted on a lot of stuff that eventually trades at a different bank, so they can often infer the identity of the client when they see the print hit the tape.
CNBC, on the other hand, know nothing and simply report what they are told and what they observe in regulatory filings. Their chief function is not to break news, but rather to distribute it -- like a buddy who is very up-to-date on current events.
TL;DR You can often deduce some of the characteristics of a position, but there are ways for a very stealthy market participant to hide the characteristics of his stake.