How does Dash achieve anonymity without centralisation? Dash is supposedly completely anonymous. How does this work without any centralisation? Is there one person with all the transactions and only they mine?
TL;DR yes, it's still decentralized, no it isn't perfectly anonymous.

From the Darkcoin (now Dash) Wikipedia: "It adds privacy to transactions by combining identical inputs from multiple users into a single transaction with several outputs. Due to the identical inputs, transactions usually cannot be directly traced, obfuscating the flow of funds." The nodes who perform this "mixing" are called Masternodes, and operate on a decentralized ledger.These nodes are trustless and do not have complete knowledge of input/output

In order to prevent Sybil attack, the masternode tier requires colateral of 1000 Dash to run. This makes the cost of attack very big. In this whitepaper is a lot more information for interested ones.