A Linux distribution is really thousands of separate programs all with a license, but we can just consider the Linux kernel first. The kernel is licensed under the GPL, which you really should read. The gist of it is that as a user of the kernel you are entitled to use the kernel however you want, to receive a copy of the kernel source code, to modify it however you want. Additionally, if you've modified the kernel in any way then you are also allowed to distribute those modifications to others.
However, this is not an unconditional license. The license can be revoked if you fail to follow a simple rule: if you distribute the kernel to anyone outside your organization then you must distribute it under the same terms as you received it. This means that if you modified the kernel, and distributed the modified version, then you must supply the source code for the modified version to whoever you've distributed it to. (If you haven't modified it, then you can simply tell them which git tag or revision to clone. That won't cost you much, but you do have to make sure your customer support people know how to answer the question, so it's probably not zero cost.)
But the key is that you're not required to distribute the software, modified or otherwise, to anyone. If you're not distributing the software, then you still gain the benefits.
Other programs with licenses that meet the Open Source Definition will be similar.
Note that nowhere in here is it required that you contribute your changes back to the upstream developers, only that you distribute those changes to your _downstream_ users. An upstream developer might need to become a downstream user in order to gain access to the changes, which they could then commit to the upstream code if they want.
That said, even if you don't distribute your modified version, you might decide that it's economically infeasible to merge your changes with every new upstream release. You might contribute those changes back to the kernel developers merely in the hope of reducing your future operational costs rather than because it's required.