 Solution

For each point $x$, choose a global section $s$ whose germ $s_x$ generates the rank-one free ${\mathcal{O}}_{X,x}$-module ${\mathcal{L}}_x$. The section $s\otimes s$ is global, and tensor products commute with taking stalks. Thereforegenerates $(\mathcal{L}{\otimes }_{{\mathcal{O}}_X}\mathcal{L}{)}_x$. Thus the tensor square is globally generated. More generally, the tensor product of globally generated line bundles is globally generated: choose a generator section for each factor at the point and tensor them. This works on every scheme, including nonreduced schemes.