Does the GPL require that source code of modified versions be posted to the public?

The GPL does not require you to release your modified version. You are free to make modifications and use them privately, without ever releasing them. This applies to organizations (including companies), too; an organization can make a modified version and use it internally without ever releasing it outside the organization.

But if you release the modified version to the public in some way, the GPL requires you to make the modified source code available to the program's users, under the GPL.

Thus, the GPL gives permission to release the modified program in certain ways, and not in other ways; but the decision of whether to release it is up to you.

More information can be found here.