No, but your device does need to be activated initially, and that requires an internet connection.
Miracast (aka "screen mirroring" on Roku) uses WiFi Direct which uses P2P (peer to peer) connections between the mirroring device and Roku device on Roku devices, and does not even need to be connected to an AP/wireless gateway, nor the internet (Miracast infrastructure/AP mode is not supported on Roku devices).