You and the party you are trading with do not need anyone else, and can work via Telegram or whatever. However you are taking on risks that could be avoided using a licensed exchange. If you do it yourself and someone rips you off... Well, you have no recourse to get your money back. So if trust is an issue, use an exchange.
I recommend to trade with people on Telegram only with people who you know personally, and you are sure they won't scam you.
On my experience Telegram is so full of scammers that I strongly suggest
against doing any trade on it with unknown people.
If you don't want to use exchanges to avoid giving your personal information, then you can choose from any exchange that doesn't require KYC (for example Cryptopia)
Edit: added bolded text