This paper proposes LinkBlackHole*, a novel algorithm for finding communities that are (i) overlapping in nodes and (ii) mixing (not separating clearly) in links. There has been a small body of work in each category, but this paper is the first one that addresses both. LinkBlackHole* is a merger of our earlier two algorithms, LinkSCAN* and BlackHole, inheriting their advantages in support of highly-mixed overlapping communities. The former is used to handle overlapping nodes, and the latter to handle mixing links in finding communities. Like LinkSCAN and its more efficient variant LinkSCAN*, this paper presents LinkBlackHole and its more efficient variant LinkBlackHole*, which reduces the number of links through random sampling. Thorough experiments show superior quality of the communities detected by LinkBlackHole* and LinkBlackHole to those detected by other state-of-the-art algorithms. In addition, LinkBlackHole* shows high resilience to the link sampling effect, and its running time scales up almost linearly with the number of links in a network.