If you buy and sell Bitcoins from a person there is no way for ANYBODY to track it. If you can ask for cash when selling and pay in cash when buying that would make thing perfect, no trace what so ever that you made a transaction! Bitcoin transactions via the blockchain is 100% anonymous, so there's no way to track that its you who are move them everywhere.
This is the best way to retain privacy -- p2p trading with cash. Technically, it isn't
anonymous since you still need to interface with the buyer/seller. He knows some things about you like Localbitcoins username, what you look like, maybe your phone number.
That's the only way I use Localbitcoins -- basically as classified ads. You won't pay escrow fees or deal with KYC that way.
The government can also use IP addresses to track you.
Use TOR for web browsing (including broadcasting raw transactions). You can also automatically route your internet connection through a good VPN like Private Internet Access.
I doubt this is really possible or practicable. For the purpose of investigating a crime, that may be reasonably feasible but not to monitor transactions for the general public.
Indeed. But let's think about that.
When you interface with the internet, web sites, merchants and third party payment processors, you leave behind an abundance of data. Cookies, IP addresses, email addresses, usernames, wallet public keys/addresses (as with lightweight wallets), browser and OS information, etc. Much of that data will be stored, somewhere, in a database.
And one day, when the police/government are investigating you, that information might become accessible to them. That may not matter, depending on what you are hiding (or not hiding). But due to the public nature of the blockchain, if they can tie some transactions to you, they may be able to discover much more about your historical activities and financial holdings.
Maybe this possibility is remote; maybe I am paranoid. But I think we all ought to be thinking about this shit now, before governments and blockchain analysis companies develop more sophisticated surveillance systems.