Although wood is not my specialty, I have investigated similar complaints
on concrete walkways. In all cases, the "wet spots" were concentrated salt
solutions.

Salt is hygroscopic-- it will absorb sufficient moisture from air,
particularly at high humidity, to get itself wet. It never entirely gives
up this moisture, but the effect is less evident at lower humidity. In a
chemical process I once worked on, it required 4 hours of 180 degree (F)
heat at full vacuum to dry salt cakes down to around 2% moisture. To keep
it from getting wet again, it had to be handled in a humidity controlled
building at less than 25% RH.

The pet urine theory doesn't conflict with this, as urine is indeed salty.
That's my guess, anyway, as to what could be going on. 

Mike Edison