The legal agreements presented when installing the software will define who owns what. Not a gut feeling.
Many put windows or MacOS on their PCs. It is present on their hard drives, and their optical media or thumb drives.
Do they own it?
Nope.
Even if you have Linux installed, you don't own it. You're licensed to use it. Some of it is GPL, some is MIT, some is LGPL, etc. All LICENSES for use. Licenses which can be revoked at any time. You don't even own Linux when you install it on the PC you own.
There are two points I would like to pick out of earlier discussions and explain my point of view on them:
1. Ownership
2. Offline
Unless I am plain wrong, you live in the US, I live in Finland. I can assure you, the OS I buy and install on my machine belongs to me. Even if the license to use the software expires, I can do "whatever the hell" I want with the bits in my private home. Don't get me wrong - I understand your point but I do not agree in this case. EULAs are different in other parts of the world because laws are different. You seem to know much more about legal issues involving ownership / IP than me, but I would like to point out that discussions like these (about who owns what) are usually very heated between EU / US due to this difference. We've seen it in other threads about hacking oscilloscopes, apparently a NO-NO in the US while in Germany it is yours to fiddle with it. (and lose warranty and what not in the process)
It does not make sense to me to argue about Altium's software while looking at All those GPL, MIT and other license agreements since they are not what is used for the software. I have not seen the actual EULA on their webpage for this coming software but I have not really been trying to find it!
About the offline thing:
I don't like how it is assumed that people's arguments against online use are moot. I never said I was going to do serious CAD in a bus or train (but maybe train, it's almost a 2 hour ride to work one direction) and we actually do have pretty good broadband or WiFi in the long distance trains, at least here in Finland. The place where I like to do CAD at home has no internet access. I chose to do that due to various reasons: I don't want the internet to distract me, I don't want that particular machine connected to the internet. I don't want to end up in a situation where I want to do layout work but the damn internet is down (I assume that it means I cannot use the software properly when offline?)
Those are things that I would have to consider when using this new software and they are of course personal reasons. I'm looking forward to giving it a try though since it is new and exciting and Altium's attempt to get into this part of the market. If I purchase the software and know that the intention is to share the design files in the cloud then me "as hobbyist/private user" may not really mind it, but if I disagree with this philosophy then I'd better not buy the product (heh, it's free is it? Nvm).
One thing is certain though, if I can't save the files and use my offline computer to access the files whenever I want (= in 5 or 10 years) then it is not the kind of software I would use.
Microsoft Office works quite well with this OneDrive cloud thing, you can use a web client from wherever you are and use Word or Excel etc. The files are in your cloud drive but also local on your home PC. You also don't have to use the cloud, i.e. files I don't want in the cloud will not be there.
EDIT: Bored@Work reminded me of World of Warcraft (an online game with subscription, you spend time to develop an avatar that you do not "own"). You can't use the service during maintenance and when something goes wrong and the data is gone, you get a wet hand shake and a pat on the back. Of course they do what they can in order to restore data (they want the subscription of course) but here the EULA and the game model actually very much state that when the lights go out, you have nothing but fond memories of pixels.