the reason one is buying a put can be either an outright short directional trade or a volatility trade which is non directional.
Something like that. Simple reason, the put buyer is buying insurance for a long stock position. There are many reasons for the opposite side. Who cares what they're thinking, you're not the put buyer. They are doing it for their reasons, you're not in their head space. Focus on your side of the trade.
Is the question your asking as to the mindset of the other side of the trade. The trade is rarely a customer cross and your contra party is most likely to be a market maker who has an appetite to trade and will most likely add the position to their inventory. They then macro hedge their book. How they hedge and what risks they are viewing is dependent on their trading style. They may simply delta hedge - not really common for a size MM today. The big three MMs are running vol books and they may hedge by trading stock(Most expensive hedge), trading other options and they may skew volatility to make certain options more/less attractive. If it's a small trade many MMs will do nothing and just continue making markets and hedge when they hit some level of risk. Most common today for liquid stock options is to trade against a correlation to an index. They may be MMs in both and trade sort of a macro risk book. Generally, size MMs trade without regard for directionality of the trade and hedge at some "unbalance" - think 5 or 10 thousand deltas. Also, keep in mind 3 parties to any trade. You buy/sell actually has OCC in the middle - you buy on the exchange, but you are actually buying from OCC. Your option is fungible and depending on name you can open/close on as many as 16 exchanges.