Random quantum circuits, in which an array of qubits is subjected to a series of randomly chosen unitary operations, have provided key insights into the dynamics of many-body quantum entanglement. Recent work has shown that interleaving the unitary operations with single-qubit measurements can drive a transition between high- and low-entanglement phases. Here we study a class of symmetric random quantum circuits with two competing types of measurement in addition to unitary dynamics. We find a rich phase diagram involving robust symmetry-protected topological, trivial and volume law entangled phases, where the transitions are hidden to expectation values of any operator and are only apparent by averaging the entanglement entropy over quantum trajectories. In the absence of unitary dynamics, we find a purely measurement-induced critical point, which maps exactly to two copies of a classical two-dimensional percolation problem. Numerical simulations indicate that this transition is a tricritical point that splits into two critical lines in the presence of arbitrarily sparse unitary dynamics with an intervening volume law entangled phase. Our results show that measurements alone are sufficient to induce criticality and logarithmic entanglement scaling, and arbitrarily sparse unitary dynamics can be sufficient to stabilize volume law entangled phases in the presence of rapid, yet competing, measurements.