Dedoimedo posted a short tutorial explaining how to resolve the issue of missing Flash Player in Linux beta Steam software distribution service store due to incompatibility between 32-bit software and 64-bit plugins

All right, a few days ago, I showed you how to install and setup the Steam beta client for Linux on Ubuntu Pangolin, the current Long Term Support (LTS) release. As you recall, there was one relatively big issue when browsing the game store. Videos were not available, because Steam thought that Flash was missing from the system. It was indeed there, but it could not find the player and use it as needed.

In this guide, we will resolve the problem. We will learn a bit about Steam client internals, we will learn how to fiddle and manually tweak Flash for Linux, although we have done this on many occasions before. And most importantly, we will restore the missing functionality to our Steam software. After me, gents.

  How to setup Flash Player in Steam Linux