Say we have two particles which are entangled so that they have opposite spins. If one is up, the other is down. They are sent off to two spatially separated observers A and B. Both observers can claim the following: The probability of measuring their particle in the spin up state is 1/2 = probability of measuring spin down state. Now, suppose observer A measures his particle in the spin up state. Now, the probability of observer B measuring spin down = 1, and probability of measuring spin up = 0. I think it is fair to say that the Observation made by A influenced the outcome measured by B. He could have gotten spin up if he had measured his particle before A. However, there is no transfer of information due to the random outcomes. B doesn't know whether he measured spin down due to A measuring up, or if he was the first to measure his spin. Observations made by A are random too, and so if we repeat this experiment many times, B will measure spin up 1/2 the time and spin down 1/2 the time. However, the ordering of which measurements report spin up and spin down will be different than if B always measured before A. The probability would still be spin up 1/2 the time, and spin down 1/2 the time.
Are we left to say that relativity is about information and not this subtle influence? Einstein certainly was not of this opinion, "The following idea characterises the relative independence of objects far apart in space (A and B): external influence on A has no direct influence on B; this is known as the Principle of Local Action, which is used consistently only in field theory. If this axiom were to be completely abolished, the idea of the existence of quasienclosed systems, and thereby the postulation of laws which can be checked empirically in the accepted sense, would become impossible." -- A. Einstein