Efficient characterization of highly entangled multi-particle systems is an outstanding challenge in quantum science. Recent developments have shown that a modest number of randomized measurements suffices to learn many properties of a quantum many-body system. However, implementing such measurements requires complete control over individual particles, which is unavailable in many experimental platforms. In this work, we present rigorous and efficient algorithms for learning quantum many-body states in systems with any given degree of control over individual particles, including when every particle is subject to the same global field and no additional ancilla particles are available. We numerically demonstrate the effectiveness of our algorithms for estimating energy densities in a $U(1)$ lattice gauge theory and classifying topological order using very limited measurement capabilities.