  Monitored quantum circuits exhibit entanglement transitions at certain
measurement rates. Such a transition separates phases characterized by how much
information an observer can learn from the measurement outcomes. We study
SU(2)-symmetric monitored quantum circuits, using exact numerics and a mapping
onto an effective statistical-mechanics model. Due to the symmetry's
non-Abelian nature, measuring qubit pairs allows for nontrivial entanglement
scaling even in the measurement-only limit. We find a transition between a
volume-law entangled phase and a critical phase whose diffusive purification
dynamics emerge from the non-Abelian symmetry. Additionally, we numerically
identify a "spin-sharpening transition." On one side is a phase in which the
measurements can efficiently identify the system's total spin quantum number;
on the other side is a phase in which measurements cannot.
