This paper presents an adaptive rate control scheme for streaming-based content sharing service. This scheme delivers multimedia contents from a user device to another device or seamlessly redirects streaming service across heterogeneous user devices. In the proposed scheme, a streaming server adjusts video quality level according to the network and client status. Our scheme is different from other rate control schemes, because the video quality at the server is decided not only based on the available bandwidth, but also based on the device characteristics and bandwidth requirement at the access network. We also propose a bandwidth estimation method to achieve more equitable bandwidth allocations among streaming flows competing for the same narrow link with different Round Trip Times (RTTs). Through the simulation, we prove that our scheme improves the network stability and the quality of streaming service by appropriately adjusting the quality of the video stream. The simulation results also demonstrate the ability of the proposed scheme in ensuring RTT-fairness while remaining throughput efficient.