How active is AncestryDNA?

Information provided in the tabulation of your matches on AncestryDNA includes the date when that client last logged in.
Looking at my top 350 matches as of February 28, 54% had most recently logged in during the month; another 15% last active during January.
26% had last logged in during 2015, 4.3% last active during 2014 and 0.6% during 2013.
Based on the sample the chances of a person with whom you newly match seeing it are about 70% within two months.
Whether the match is significant enough for them to respond, and whether they will, is another matter.