There are several natural measures that can be placed on the amoeba of an algebraic curve in the complex projective plane. Passare and Rullgard proved that the total mass of the Lebesgue measure on an amoeba of a curve of degree d is bounded above by π²d²/2, by comparing it to another Monge-Ampère type measure, which is dual to the usual measure on the Newton polytope of the defining polynomial via the Legendre transform. Mikhalkin generalizes this upper bound to half-dimensional complete intersections in higher dimensions, by considering another measure supported on their amoebas; their multivolume. The goal of this talk will be to discuss these measures in the setting of random plane curves. In particular, I'll first present our results with Bayraktar, showing that the expected multiarea of the amoeba of a random Kostlan degree d curve is equal to π²d. For Lebesgue measure, it turns out that the expected asymptotics are much lower: I'll describe our results with Welschinger, showing that the expected Lebesgue area of the amoeba of a random Kostlan degree d curve is of the order ln(d)².