How can we accurately estimate local triangles for all nodes in simple and multigraph streams? Local triangle counting in a graph stream is one of the most fundamental tasks in graph mining with important applications including anomaly detection and social network analysis. Although there have been several local triangle counting methods in a graph stream, their estimation has a large variance which results in low accuracy, and they do not consider multigraph streams which have duplicate edges. In this paper, we propose FURL, an accurate local triangle counting method for simple and multigraph streams. FURL improves the accuracy by reducing a variance through biased estimation and handles duplicate edges for multigraph streams. Also, FURL handles a stream of any size by using a fixed amount of memory. Experimental results show that FURL outperforms the state-of-the-art method in accuracy and performs well in multigraph streams. In addition, we report interesting patterns discovered from real graphs by FURL, which include unusual local structures in a user communication network and a core-periphery structure in the Web.