Our modern network streaming devices will require you to to set the device up on a network before it can be used in most applications.
However, connection to the internet itself is not required - just connection to a network.
A network is a group of devices (like phones, computers, or printers) that are connected to each other, allowing them to share information.
An internet connection, on the other hand, is the link that lets your devices access the internet, a massive system of networks all over the world. This connection lets you browse websites, send emails, and use online services.
You don't need to be connected to the internet - but you will need to be on a network so that your phone or tablet can control the network streamer and music playback.
You could, if needed, use a simple router without connecting it to the (coaxial) connection that usually connects to a cable or service provider system for internet access. This will still put out a wi-fi signal you can connect to, but won't connect you to the web.