We consider the joint value distribution of Dirichlet $L$-functions in the critical strip $\frac{1}{2} < σ< 1$. We show that the values of distinct Dirichlet $L$-functions are dependent in the sense that they do not behave like independently distributed random variables and they prevent each other from obtaining large values. Nevertheless, we show that distinct Dirichlet $L$-functions can achieve large values simultaneously infinitely often.