Anything said here will obviously not be legal advice.
Most extensions use the MIT license without any special provision, so any documentation online about MIT license will apply as well.
ChooseALicense website has some nice summary https://choosealicense.com/licenses/mit/
If you're using a modified version of an extension privately, you have no obligations to publish your code. This is different from some other popular open-source licenses.
You should however always retain the copyright notice of the original extension in your source code.
If you decide you want to re-distribute/open-source your code, you need to include the original copyright notice and the license you choose for your new extension needs to be compatible with the original license. Often forked extensions will also use MIT and just add a new author to the list in the license file, while keeping the original author(s) and license text.
There is generally no obligation to say your website is using a given MIT extension or library, the copyright notice is only for re-distributing code, not for offering a service online.
In the Flarum ecosystems there are also ways to extend/override extensions without copy-pasting any of their code, in which case you aren't bound to their license in any way.
