You might be right. In the Widex Technology link in the OP, there are the following lines:
" The only waste from this reaction is vaporised water and negligible amounts of carbon dioxide."
and for recharging:
"Simply place the hearing aid into the refill well and wait for 20 seconds.
In that time, the refill unit will extract any remaining methanol and supply the fuel cell with new methanol. "
But the anode reaction always needs water (see Direct methanol fuel cell - Wikipedia), so that’s why I’m presuming when the unused methanol gets flushed, there is some amount of water in it. I think it says in the Wikipedia article, too, that the requirement for water as an ANODE reactant in the reaction cuts down on the available energy density from methanol, which otherwise has something like 3x the energy density of a Li-ion battery, if I remember correctly.