At IAD (as with all US airports IIRC) arriving and departing pax can mix freely, with the exception of international arriving pax who need to be checked out by INS (and possibly security screened if they're connecting onto another flight or have to go airside to get landside)

When you land at IAD, you'll walk off the plane and end up airside, clear of security etc. You then just need to walk to your gate. Think if you were doing a domestic to domestic connection at LHR T1 (I know, you just wouldn't do it) - it's exactly the same as that.