TL;DR it depends on the use case, but I'd expect everyone in the top 5-8 cloud providers to own and operate at least some of their own facilities. But everyone is going to have some space in a collocation facility.

There are various degrees of "operate your own datacenter" in the datacenter market.

A lot of facilities on this list, like Equinix, are retail datacenters. They have things like 24/7 "smart hands" who you can call to go repair a server, replace or move a networking cable, things like that. These are the most full service options. They also offer various network connectivity products, and if you are willing to pay for it, you could go as far as ship them servers or a rack and have them do much of the install for you. These are the most expensive options, but, at least in the case of Equinix they also typically have the best network connectivity. This is your typical collocation, or even carrier hotel.

So if you go into an Equinix facility in a lot of markets you will see everyone with racks or cages full of equipment in there. Though the biggest players are mostly going to use them for network POPs and maybe something like a CDN node. So you might see Amazon CloudFront equipment in there, but not much else.

Often the thing to do is put your public internet connectivity, peering, and perhaps backbone links in a well connected place like this, but the bulk of your compute is in a much larger and cheaper facility elsewhere.

There are also retail collocation facilities that aren't super well connected (when I say well connected I mean you can get access to 10s or 100s of providers just by being in that facility) and those are a lot cheaper than Equinix, and often full service. They tend to cater to smaller customers. Though I've seen Akamai have a ton of racks in some surprisingly crappy faculties.

Then DRT (Digital Reality Trust) operates more in the wholesale datacenter market. They can rent cages for multiple racks, but are even more used to renting out whole datahalls, so you have dedicated space. In a lot of areas they can't really offer things like 24/7 smart hands. They aren't going to handhold you nearly as much, and space and power tends to be a lot cheaper, but your minimum buy is going to be a lot larger. 20-100 racks maybe?

CoreSite kind of straddles the line between Retail and Wholesale. They offer a lot of full service stuff, but can also rent you out a whole data hall or even building at a campus if you need it. No one would ever confirm this, but I'm pretty sure one of the big 3 cloud providers has a big chunk of, the CoreSite Silicon Valley campus. If you search Google Maps for core site silicon valley you'll see a cluster of about 5 buildings, all labeled. One building right there is suspiciously unlabeled, but if you do the the Satellite view, it's clearly a datacenter.

Take it another step, and you have companies like Sabey who offer a datacenter shell. Basically a building with power, generators, etc, but you do everything else, or they customize it to your specs and help operate it. Yet again cheaper per unit of compute you are doing, but a lot high minimum spend to make sense. You'll probably be measuring things in MW (megawatts) before you are looking at something like this. Sabey also sells to retail datacenters. I spent far too much time at the Sabey Tukilla campus south of Seattle from 2009 - 2011 in an Internap facility. I had to go through Sabey security to get into the building, and then Internap security to get into their space. Most everything inside that space was taken care of by Internap (now seemingly branded INAP).

Then you have build and operate your own datacenter, which also has various degrees. I don't know how it works now, but I know a few years back AWS had (has) a TON of buildings in the Ashburn Virginia area that are us-east-1. I was told by someone who worked in datacenter real estate that they typically would buy the land and get a datacenter shell built, then sell it to another operator and lease it back. Then bring in all their own power and compute.

I also heard that a few years ago when Oracle was building out their cloud, they were just buying up all the retail datacenter space they could so they could get to market faster. They were paying an arm and a leg for it, but that was the fastest way to get online.

So the economics of all this depend on what you are doing, why you have the space, and how much actual equipment you need in there. At smaller scales, say a CDN, or just networking equipment, it often makes the most sense to just lease retail space. At larger and larger scales it makes more and more sense to in-house more of it.

I've never worked at one of the large cloud providers, but I'd expect they have some smaller regions that are retail or wholesale datacenters operated by another provider. But the large regions are mostly owned and operated.