Large language model (LLM) agents increasingly act on a user's behalf -- reading personal files, calling tools, transacting with external services -- possibly leaking personally identifiable information (PII) across trust boundaries at every step. Privacy here is a property not of a single output but of an entire trajectory, and three properties make it hard: leakage is cumulative, as individually innocuous releases accumulate across honest-but-curious or colluding sinks into inferences about a protected secret; bidirectional, as a malicious observation can inject instructions that turn the agent's own reasoning model against the user; and task-dependent, as the same field is necessary for one recipient yet gratuitous for another. Per-release contextual-integrity filters, information-flow controls, and posterior-leakage monitors each address part of this but none controls cumulative, inference-based leakage at runtime. We recast agent privacy as \emph{posterior-risk control} and present OCELOT, a runtime mediator that budgets how much an adversary's belief about a secret may improve across a trajectory, rather than filtering outputs. Its mechanism, \emph{Witness-Verified Declassification}, separates judgment from trust: an untrusted, locally fine-tuned defender model inspects each candidate release and emits structured evidence -- labeled atoms and proposed declassification operators -- which a deterministic verifier audits, charging a certified min-entropy cost for the chosen variant and authorizing the least-disclosing useful release under a sink-trust-weighted budget recorded on a tamper-evident ledger. Across diverse agent benchmarks and recent defenses, OCELOT attains significantly lower leakage at higher task utility, resists adaptive injection, jailbreak, cumulative inference, and sink collusion, and adds only modest overhead.