We study conserved stochastic sandpiles (CSSs), which exhibit an active-absorbing phase transition upon tuning density ρ. We demonstrate that a broad class of CSSs possesses a remarkable hydrodynamic structure: There is an Einstein relation σ^{2}(ρ)=χ(ρ)/D(ρ), which connects bulk-diffusion coefficient D(ρ), conductivity χ(ρ), and mass fluctuation, or scaled variance of subsystem mass, σ^{2}(ρ). Consequently, density large-deviations are governed by an equilibrium-like chemical potential μ(ρ)∼lna(ρ), where a(ρ) is the activity in the system. By using the above hydrodynamics, we derive two scaling relations: As Δ=(ρ-ρ_{c})→0^{+}, ρ_{c} being the critical density, (i) the mass fluctuation σ^{2}(ρ)∼Δ^{1-δ} with δ=0 and (ii) the dynamical exponent z=2+(β-1)/ν_{⊥}, expressed in terms of two static exponents β and ν_{⊥} for activity a(ρ)∼Δ^{β} and correlation length ξ∼Δ^{-ν_{⊥}}, respectively. Our results imply that conserved Manna sandpile, a well studied variant of the CSS, belongs to a distinct universality-not that of directed percolation (DP), which, without any conservation law as such, does not obey scaling relation (ii).