Most open source licenses allow you to run the program for any purpose, and only require you to reproduce the license or copyright statement when you pass on a copy of the program, whether in source or binary form. The details depend on the license.
When you run the open-source software only on your servers and do not give a copy of your software to other legal entities, these licenses do not impose any restriction. Concrete examples:
The GPLv2: “Activities other than copying, distribution and modification are not covered by this License; they are outside its scope. The act of running the Program is not restricted”.
The Apache v2 License does likewise not restrict your ability to run the program. It also gives you a copyright license to “publicly perform” the program, and a patent license to use the work.
So these licenses do not oblige you to display any notices about the open-source components used by your server-side code.
An open-source license that is notably different is the AGPLv3. It “explicitly affirms your unlimited permission to run the unmodified Program” and to “make, run and propagate covered works that you do not convey”. However: “if you modify the Program, your modified version must prominently offer all users interacting with it remotely through a computer network (if your version supports such interaction) an opportunity to receive the Corresponding Source”. If your system has a client side user interface, the user interface can display these acknowledgements together with other legal notices. In other cases, this is more tricky and should be covered in another question.