This paper studies the joint tail behavior of two randomly weighted sums ∑_(i=1)^m Θ_i X_i and ∑_(i=1)^mθ_j Y_j for some m, n∈N∪{∞}, in which the primary random variables {X_i;i∈N} and {Y_i;i∈N}, respectively, are real-valued, dependent and heavy-tailed, while the random weights {Θ_i,θ_i; i∈N} are nonnegative and arbitrarily dependent, but the three sequences {X_i;i∈N}, {Y_i;i∈N} and {Θ_i,θ_i; i∈N} are mutually independent. Under two types of weak dependence assumptions on the heavy-tailed primary random variables and some mild moment conditions on the random weights, we establish some (uniformly) asymptotic formulas for the joint tail probability of the two randomly weighted sums, expressing the insensitivity with respect to the underlying weak dependence structures. As applications, we consider both discrete-time and continuous-time insurance risk models, and obtain some asymptotic results for ruin probabilities.