Actually, p-adic L-functions are expected to satisfy functional equations compatible with the classical ones. For M an ordinary motive, Coates and Perrin-Riou conjectured the interpolation property at critical integers and the expected functional equation in some papers in the early nineties (see for example thisthis). In particular, the Kubota-Leopoldt p-adic L-functions interpolate all critical values of the classical Dirichlet L-functions (up to a period and a multiple). For modular forms, Mazur-Tate-Teitelbaum, in their 1986 paper) prove a p-adic functional equation in section 17. In fact, the two-variable p-adic L-function of an ordinary family of modular forms satisfies a two-variable functional equation interpolating the one-variable functional equation at each weight (see for example Greenberg-Stevens' inventiones paper) (I'd post more mathscinet links but it appears to be down...).
As for the values of the p-adic L-function at non-critical integers, that's much more mysterious. Rubin has a computation outside of the critical points for a CM elliptic curve in section 3.3 of his paper in the "p-adic monodromy and BSD" proceedings. I think I've seen other cases, but generally it takes a lot of effort, I think.
(Also, regarding Iwasawa theory's concern with values of L-functions, it is true that the Main Conjecture is only an equality of ideals in some power series ring, but one can still hope to construct p-adic L-functions on the analytic side that do a nice job at interpolating, say up to a p-adic unit.)
