When I read about, say, some hedge fund wanting people with such-and-such Sharpe ratio, how is that ratio recorded and communicated to the interested party? I mean, do people just take it on faith that if I say my Sharpe ratio is X, is it so? Or is there some formal process where this can be recorded?
The hedge fund will usually ask people to provide a time-series of their strategy's returns, preferable from live trading and preferably at a daily frequency. Sometimes they will accept a mix of backtested and live results, so long as they are not too different from each other, both in theory and in practice. Some funds break down their monthly returns by strategy, and people who have worked at such funds in the past can provide this information and indicate which strategy was their responsibility, but then it will be taken "on faith" that this person was, in fact, the primary person responsible for a given strategy.
Bottom line, I do not think there is any formal process for recording this information, and the standards of the potential employer/investor will vary widely.
The GIPS standards are increasingly used for presenting investment returns in a standardized fashion across equities, real estate, private equity, fixed income and other asset classes.
The GIPS standards rely on, among other things, chain-linking time-weighted returns and they require specific disclosures including carve-outs, net or gross performance, treatment of discretionary accounts, and definitions of composites and benchmarks.
There is no specific guidance on Sharpe ratio which is a shame since the ratio can be gamed by trading illiquid (serially correlated assets), by extending the measurement horizon, or engaging in a return swap where you pay out both the best and worst performing months to a dealer.
However, by presenting the returns in a GIPS compliant fashion an investor can evaluate against performance metrics of their choosing.
A firm can also verify GIPS compliance by an independent third party.