For a nontrivial graph $G$, its first and second Zagreb coindices are defined, respectively, as $\overline{M}1(G)=\sum_{uv \notin E(G)}(d_G(u)+d_G(v))$ and $\overline{M}_2(G)=\sum_{uv \notin E(G)}d_G(u)d_G(v)$, where $d_G(x)$ is the degree of vertex $x$ in $G$. We explore further properties of Zagreb coindices. First, we investigate Zagreb coindices of two classes of composite graphs, namely, Mycielski graph and edge corona, and we present explicit formulas for Zagreb coindices of these two composite graphs. Then we we give two estimations on Zagreb coindices of graphs in terms of the number of pendent vertices and Merrifield-Simmons index, respectively. Finally, we give several Nordhaus-Gaddum type bounds for the first Zagreb coindex.