Triangle counting in networks under LDP (Local Differential Privacy) is a\nfundamental task for analyzing connection patterns or calculating a clustering\ncoefficient while strongly protecting sensitive friendships from a central\nserver. In particular, a recent study proposes an algorithm for this task that\nuses two rounds of interaction between users and the server to significantly\nreduce estimation error. However, this algorithm suffers from a prohibitively\nhigh communication cost due to a large noisy graph each user needs to download.\n In this work, we propose triangle counting algorithms under LDP with a small\nestimation error and communication cost. We first propose two-rounds algorithms\nconsisting of edge sampling and carefully selecting edges each user downloads\nso that the estimation error is small. Then we propose a double clipping\ntechnique, which clips the number of edges and then the number of noisy\ntriangles, to significantly reduce the sensitivity of each user's query.\nThrough comprehensive evaluation, we show that our algorithms dramatically\nreduce the communication cost of the existing algorithm, e.g., from 6 hours to\n8 seconds or less at a 20 Mbps download rate, while keeping a small estimation\nerror.\n