The SpaceX ground stations are meant to be consumer owned (about the size of a pizza box).
On the other hand it is completely possible that SpaceX's ground stations will use HW that is similar to their CPE box and thus they can build ridiculous amount of such ground stations (ie. at every IXP or so, althought the "lightweith non-IP transport" proclamation somehow precludes doing that)
[1] https://motherboard.vice.com/en_us/article/d34bmk/iridium-ne...