I had a device which did this very well between my two laptops. The most common solution to this problem is a Keyboard-Video-Mouse (KVM) switch, but because my laptops already have individual screens, the "V" in "KVM" wasn't necessary. That's why I bought a "Keyboard-Mouse" switch, AKA a "KM" switch.
Unfortunately, with the way Google's search engine handles keywords, you will usually see a lot of "KVM Switch" results when you search for a "KM Switch" - this makes these pretty hard to find for people like us, as well as pretty hard to sell for people who make these - which is why they cost so much.
The device I purchased for around $150 in 2015 was the Avocent/Emerson SVKM120, which switches USB mouse and keyboards between two computers. It always worked reliably and was super useful. I sold it, which was dumb, because in 2019 those things are going for like $235 on Amazon. The 4-computer version is currently $370. There's also the Belkin KM switches, but they are even more expensive. The prices are probably inflated because the US military uses a nearly unlimited budget to buy, in addition to $600 toilet seats, these - at whatever price. The military has a lot of people who have to access/administer computers on separate networks that can't be connected - these devices are surprisingly useful for doing exactly that.
Alternatively, if you are able to connect your computers on a network, you would use the software solution Synergy to control all of them using the keyboard/mouse connected to one of them. Synergy does the job for many people. It's advantages over a KM switch are lower cost, cross-platform support, and the flexibility that comes with software. Potential disadvantages are it requires that your computers all run Synergy and that they be on the same network. Especially in corporate/gov environments, these are kind of deal breakers.
Another option: if you have only Windows machines to worry about, and have sufficient access to install/run software, then another solution is the various USB-to-USB cables that allow one computer to control the other, and vice-versa. I use one of these at work to share the mouse/keyboard on one computer with the other. There are other makes/models out there, but the one I use is the USB 3.0 version of the StarTech.com KM Switch Cable. It's not as reliable as the Avocent/Emerson SVKM120, but it's also a fraction of the price, and doesn't require network connectivity either.
I wish there was an open-source solution to this problem because the supply/demand conditions for this technology don't really work in our favor...
For example, I have some split-keyboards running QMK. I wish I could connect each half to a different computer, then connect both halves, and control which computer the split keyboard outputs to. Something similar is possible now with how bluetooth-enabled QMK keyboards can switch between wired and bluetooth connectivity, but that then makes you depend on bluetooth, which has it's own quirks... far from perfect.
Another project in this space, which may or may not be dead, was keyplus. This works kind of like the mitosis firmware did, where the keyboard has wireless transmitters that a receiver device plugged into a computer interprets. The keyboard is wireless, but the computer doesn't realize it. There was a video posted (can't find the link) where the person behind the firmware project was able to switch his keyboard & mouse between receivers connected to different computers. Basically, this person was able to achieve what you're talking about without using any wires. It was super awesome, but the project isn't really active, so good luck finding help to re-create it on your own. :/
One more project, which also isn't active, is the "Arduino K(V)M Switch" - basically someone figured out how to achieve what you're talking about by having one Arduino send PS/2 keyboard & mouse communication to however many other Arduinos over I2C, and those other Arduinos would emulate keyboard/mouse for the computers. It's a really cool, but depends on PS/2 (which I like, but is kind of rare in terms of support) and also unfortunately the project has been inactive/incomplete since 2016. But if it is possible to achieve this with just some Arduinos communicating over I2C, then I wonder if the split-keyboard/QMK idea I mentioned might one day be feasible - at the very least, the main QMK devs will have to agree on how to implement I2C, which is a little mixed up in it's current state.
