How would anyone know if those specific coins were used in any illegal activity?
Cause the blockchain is completely transparent.
If a hacker breaches a centralizes exchange and transfers Bitcoin from the coldwallet of the exchange into an address the had created, this particular transaction can be tracked as well as the wallet the inputs were sent to, from then on, any other transactions those inputs were used in can also be monitored.
You can do this yourself by visiting a blockchain explorer like blockchair.com
If I purchase bitcoins via p2p, can I get some problems if the bitcoins I get were involved in some criminal activity ? If yes then how do I avoid such situation ?
• Theoretically, you should not get into any trouble as long as you can prove that you had nothing to do with it, with orovesof transaction and communication with the other party.
• How to avoid such situation would be by keeping records if your transactions should they be needed.
Also, do not trade directly (physically) with someone except you explicitly trust this person.