Specifically talking about the lead-acid battery, I understand (or think I understand) this much:
The anode, in the presence of the electrolyte, $\ce{H2SO4}$ will be oxidised:
$$\ce{Pb (s) + SO4^2- (aq) -> PbSO4 (s) + 2e-}$$
At the same time at the $\ce{PbO2}$ electrode, two electrons from the $\ce{PbO2}$ solid are lost to the electrolyte: $$\ce{PbO2 (s) + 2e- + 4H+ (aq) + SO4^2-(aq) -> PbSO4 (s) + 2H2O}$$
This all occurs without a solid conductive path (say a wire) between the anode and cathode, the cathode will build a positive charge while the anode builds a negative charge and ions will diffuse appropriately during this process.
When a wire is connected between the anode and cathode, electrons that have built up on the surface of the $\ce{Pb}$ will migrate to the $\ce{PbO2}$ which is of lower potential and if a load is placed along the way work can be done. So batteries will discharge by themselves without even contributing to any work? Is this not true for all batteries?
The part that confuses me is that most people say that batteries store chemical energy.
I understand that the redox reactions release energy which can be used as electrical energy but if batteries discharge without contributing to any meaningful work then why are they still considered to be storing energy? Aren't they converting chemical energy to heat continually unless they are put to use?