Path, of course, knows where I am due to the geolocation of the IP from which I post.
My intent was communicated to the app through the disabling of location services. Posting my location uncovered through parsing EXIF is the opposite of my configured intent.
When I post pictures to Twitter, Twitter receives the EXIF-tagged photos, too. They, however, don't serve them with the geotags, as I have disabled geotagging for my account. I haven't tested if they just strip all photo EXIF, or only for accounts that disabled geotagged tweets.