One Laptop - 2 Domains

You can only be joined to one domain at a time. However, there are workarounds for him to be able to access resources on both systems like remote desktop connections, VPN, etc.

The first question here is what do they need to access on each domain?

Access to printers, documents and applications is what is required.

The user can log into both domains, but the laptop itself can only belong to one or the other (or neither).

One Windows installation can only be a member of one Windows Domain.  You cannot join it to a second domain.

From a security standpoint, you shouldn't be connecting personal computers to your domain. It's risky, especially since they could get infected with something on their how network (or even on the other school's network) and spread it to your network.  

If you insist on doing it (vs. giving the teacher her own computer from the school), there are only two ways that come to mind:
1. VM
2. Dual boot

With the VM solution, she continues to leave the computer on the existing network, but you enable Client Hyper-V on the computer and install Windows into a virtual computer.  This effectively gives her a second computer on her existing laptop which in turn allows her to work off that computer.  The problem is it's not terribly easy for a non-technical person to start the machine and know to work in it and not the main one.  You also need to get the appropriate licenses for her (I believe that VM solution will technically require VDA/VDI licenses and those are only available from Windows 10 Enterprise.

With the Dual Boot solution, you would do a parallel install of Windows and she would be prompted on boot into one installation or the other.  It would be highly recommended to use partitions as William Fulks mentions, but I don't think it's required (I haven't set up a dual bool system in close to 15 years because I use VMs).  If you went this way, the second Windows would need to be installed into a folder like C:\win10.  The problem is program files folders would be "merged" and especially if they weren't both up to date, I can see it creating problems.  On different partitions, no name changes are required and nothing gets mixed up.  This is also the easiest for the user to work with.  When the computer is turned on a menu is presented - the select the installation they want to start and that's it.  Nothing more to do; the computer starts and they can work.

The last option is to just connect it to the network.  When a computer is NOT part of your domain, it works like a Windows 10 home computer. She'll need to specify user name and password to connect to network resources.

The only way a laptop can be connected to multiple domains if they are related or have a trust relationship.

Why not issue a laptop/co mk puter...

Seriously, though. If they need access to resources on both domains, then both employers should provide a computer to them.