What is the cheapest fee based bitcoin / altcoin mixing service to keep myself anonymous and from being tracked? I purchased some bitcoins from Coinbase and I want to give them away to friends for their birthday this year, but what I'm afraid of is if they end up buying stuff on the dark web with it. Last thing I want to deal with is having to explain to the police it wasn't me. I know there a bunch of mixing services to help protect people from being tracked or traced back to the original account, but which one is has the cheapest fee? 

I know the fee pays for the person running the mixing service, but is there such a thing as a decentralized bitcoin mixing service that doesn't take a fee?
Even if you use a mixing service it is not guaranteed that the link between the two+ addresses will disappear. Actually, as far as I know it cannot be guaranteed by any mixing service, decentralized or not, including extra fees or not...

In theory, a good mixing scenario would use a purely decentralized service with a consistently huge amount of transactions, which will randomize the mixing better. None of the services I have checked a couple of months ago had both of the above requirements. Personally I believe that doing a non-perfect mixing is worse than doing any mixing since they will know that you tried to "hide" the relationship between the addresses.

Looking around again I found this which has some good advice: in summary, to use mixing services behind the Tor network and send the bitcoins to wallet addresses also created from the Tor network. You can then check if there are any relationships remaining between your original addresses and the new addresses.

By the way, I know a lot of people that would be happy to pay a fee for a proper mixing service.