You may not get the exact street address, but you could definitely figure the block or the street. Maybe not super useful in a dense city, but that could leak quite a bit about you if you live in a suburb or rural area.
And ride / run through the circles on occasion. These will show up in tracing confusing the analysis.
They do offer manual route trimming from both ends, too. I've found this useful mostly when I forget to turn off my watch after running.
I wish they had a way to do an auto-trim so my bike rides didn’t include GPS bounce leaving the building. CommuteMarker.com had auto-tagging for Strava rides but not auto-trimming last time I looked.
For instance, if I set a privacy zone centered on my house (or my military base) with a radius of 100 miles and all of my running activities are contained within that privacy zone are those runs automatically marked private? If I never run across the boundary then there's no issue of inadvertently exposing the privacy zone center point. Would Strava hide the GPS information but allow my run stats to accumulate? I don't think the feature is quite there yet but it could be close.