Many emerging applications of AI--from scientific discovery to medical diagnosis--require agents to seek information strategically: forming hypotheses, asking targeted questions, and making decisions under uncertainty. In high-stakes settings with limited resources, do language models (LMs) behave like rational agents? Drawing on insights from human cognition, we develop methods to evaluate and enhance agentic information-seeking. First, we introduce a decision-oriented dialogue task called Collaborative Battleship, in which a Captain must balance exploration (asking questions) and action (taking shots), while a Spotter must supply accurate, contextually-grounded answers. Compared to human players (N=42), we find that many LM agents struggle to ask informative questions, produce accurate answers, and identify high-utility actions. To address these gaps, we develop novel Monte Carlo inference strategies for LMs inspired by Bayesian Experimental Design (BED). For Spotter agents, our approach boosts accuracy by up to 14.7% absolute over LM-only baselines; for Captain agents, it raises expected information gain (EIG) by up to 0.227 bits (94.2% of the achievable noise ceiling). Combined, these components yield sharper targeting (+0.303-0.374 F1), and enable weaker LMs, such as Llama-4-Scout, to outperform both humans (8% -> 82% win rate) and frontier models (0% -> 67% win rate vs. GPT-5) at ~1% of GPT-5's cost. We replicate these findings on Guess Who?, where our methods significantly boost accuracy (+28.3-42.4 p.p.), demonstrating their general applicability for building information-seeking agents.