When on foot, I'm not using the Mapillary app but an alternative that allows me to switch applications and even turn off the screen while it continues shooting. Most of the time I even have the phone cover closed.
I am casually holding my phone at hip height, carefully pointing the camera somewhat upwards to get whole buildings in the frame. In urban environments I am mostly photographing sideways instead of forwards, because I think in such environments getting the building fronts is most interesting. This way, nobody notices anything, and I can actually walk in a good pace while making photos. The photo quality is not the best (often somewhat lopsided because I can't see the screen, and cars are often blocking the view because I'm holding the phone at hip height) but I'd rather walk a street several times than make it so obvious that I'm taking photos.
Photographing forwards is a bit more conspicuous, holding my phone in front of my belly, but I learn that people only notice it if I am feeling awkward myself. If I do it casually, then nobody pays any attention.
In public transport I often glue my phone to the side window using some blu-tacks. This attracts more attention, but funnily enough people are more interested in how I get my phone to stick to the window, than into what I am actually doing.
Note that this is in Norway, where smartphones are very common, and society is so safe that people don't feel threatened by weird behavior.