The underlying technology is broadly the same for all sensor makers and is the same technology used to produce other chips (such as CPU's). Manufacturers tend to be coy about where a chip was made (fabbed) and although a teardown may yield the information it is not guaranteed. Much of the information in the public domain comes from 'teardowns' or financial disclosures on the sale of fabs by photographic companies.
There are 3 ways for camera makers to go. Some major manufacturers have their own foundry to make their chips in-house, some design chips and they are made (fabbed) to order, and some just buy in a package designed and fabbed by a third party and don't get their hands dirty with the sensor design at all.
But the whole thing is a big grey area, and it's worth bearing in mind we're working with tolerances on the µm scale, so if a third party is involved then they'll definitely be producing a product to their clients specifications. I've included a handful of the major players and what I could find out so far.
Canon state that they produce all their EOS sensors in-house but they have been on quite a large/old process (the biggest drawback of in-house production). They were rumoured to be opening a new fab at some point in 2014 and it now appears to be online. In the release they've been very careful to discuss only their EOS sensors, suggesting other types of camera may be using outsourced parts.
Fuji seem to now be using Sony and Toshiba chips, but have a differently patterned CFA for the EXR branded sensors. Rumours are they sold their fab to Toshiba.
Nikon do not have a fab facility (that I'm aware of or could find) so no sensors are produced in-house. They are known to use sensor packages from Sony, although it is likely that they had a big hand in the development of those chips. Also they use TSMC/Renesas to directly produce Nikon designed chips. Nikon seem to be fairly 'promiscuous' and do use other companies too (like Micron/Aptina, and Toshiba in the case of the D5200.) so this list is unlikely to be exhaustive and it's not unreasonable to expect that each model (or even each production run) may even have sensors fabbed a different provider.
Producing a list of sensor sources for Nikon bodies would be quite a big task, and if someone has produced one I've not come across it.
Olympus seem to buy in sensors outright from various sources.
Pentax had a strategic partnership with Samsung, but seem to use Sony now (and some Kodak in medium format.)
Samsung produce sensors and are a fairly prolific foundry in their own right, producing sensors for even arch-rivals in the consumer space (including the iPhone).
Sigma appear to have purchased the 3rd party fab (Dongbu HiTek) that they used previously to produce their Foveon sensors though they are operating it as a separate entity.
Sony are likely produce their top-end sensors in-house but as a cost cutting measure they have outsourced some production work to Fujitsu in 2010 and probably still do.
It just leaves the question: what do you want/think this information would tell you if you had more of it? Realistically the information is only really useful for specification based willy-waving.
What makes the image is the entire image formation and processing pathway. After all, an image is the result of a whole system, not merely the sensor. It's an important part but things like the selection of high speed amplifier, A/D converters, and processing in the digital realm all play their small but equally important parts.
Extra: Iliah commented that Image Sensors World contains a good deal of useful information from the sensor production side rather than brands as we see them but if you're really interested then there's lots to find there.
Is it public knowledge what manufacturers make their own sensors and what import it from others?
For some and not others.
Knowing it is nice to know and fun from an industry-watching perspective, but the results from the system as a whole are more important. Once you factor in a lens the potential improvement that knowing/using these specs and what they can offer are so small as to not be worth worrying about.