Electrical conductivity of organic solvents
Ethanol and methanol dissociate into ions, just as water does.  For water:

$$\ce{2 H2O &lt;=&gt; H3O+ + OH-}$$

For an alcohol $\ce{ROH}$:

$$\ce{2 ROH &lt;=&gt; ROH2+ + ^-OR}$$

For water the self-ionization constant is $K_\mathrm{w} \approx 10^{-14}$.  For ethanol or methanol, I was unable to find data on the self-ionization constant, which is essentially the same as the $\mathrm{p}K_\mathrm{a}$ of the alcohol in a solution of itself.  The $\mathrm{p}K_\mathrm{a}$ of ethanol in water is slightly higher than the $\mathrm{p}K_\mathrm{a}$ of water in water.  Also, the lower dielectric constant in alcohols will further disfavor self-ionization relative to water.  So for methanol and ethanol the self-dissociation into ions will be weaker than in water.

These ions will still be present, though, and will conduct electricity even in pure ethanol and methanol solutions, but the conductivities will be very low.  Very pure water has a resistivity of $\pu{18.2 MΩ cm}$.  If they self-ionize to a lesser extent than water does, the resistivities of methanol and ethanol will be even higher than water's.