When are genital warts contagious?

According to cdc.gov and dr. hunter h. handsfield answer on medhelp as well as wikipedia, most HPV infections will be cleared up eventually by our own immune system.

Based on DNA PCR tests (that aren't available commercially, only for scientific journal purposes) most people who had HPV will have negative tests 2 years afterward. Negative tests mean they are in dormant state. When i say most, they are 90% though, so the 10% will need 3-5 years or even more. Several people, like 1% or less, can't cure their HPV even after 10 years. Usually this is the type of people who gets Cervical cancer from specific HPV strains (but keep in mind cervical cancer HPV strain and genital wart HPV strain aren't the same)

So to answer your question, genital wart is very contagious. Get it treated and don't have sex until you have been wart-free for 6 months. Usually warts will recur every few weeks to few months. One day it'll stop recurring though when your antibody has taken care of the virus (see my 2nd paragraph above). So make sure to make a journal about your genital wart outbreak time, and if you have been 6-8 months wart-free from the latest recurrence, you can consider yourself cleared up and you will be no longer contagious.
Genital warts are fairly contagious.