I try to ask if there is anyone around. If not, I will go for it if the missing bridge is not too far down the road. "Bridge Out 1 Mile Ahead. Local Traffic Only"
One time in PA the sign was several miles away. I went for it anyway, thinking "It's probably passable by bike." When I got to the bridge, the demo crew was literally removing the last remaining girder. Was told the detour was loooong and that if I wanted to wait they would let me cross the creek. So I waited about 30 min. Took off the cycling shoes and tossed to them to the opposite bank. Put on the flip flops. Lifted up the bike about waist high and walked across the creek at the shallowest point I could find, which was abut 2'. The crew watched the whole thing. I felt tlike they were waiting for me to wipe out, but I didn't.
Another time in PA I was faced with a long detour until I asked some locals working at a car dealership where the detour started. They told me of an easy workaround. It involved what turned out to be a benign dirt road. I guess PennDOT didn't want traffic going that way because of potential road damage. Also, it was not suitable for trucks. (The closed bridge was on U.S. 6.) There was a little traffic, so the locals were clearly using the dirt alternative.