Trusted Execution Environments (TEEs) are designed to protect the privacy and integrity of data in use. They enable secure data processing and sharing in peer-to-peer networks, such as vehicular ad hoc networks of autonomous vehicles, without compromising confidentiality. In these networks, nodes must establish mutual trust to collaborate securely. TEEs can achieve this through remote attestation, where a prover presents evidence of its trustworthiness to a verifier, which then decides whether or not to trust the prover. However, a naive peer-to-peer attestation approach, where every TEE directly attests every other TEE, results in quadratic communication overhead. This is inefficient in dynamic environments, where nodes frequently join and leave the network. To address this, we present Careful Whisper, a gossip-based protocol that disseminates trust efficiently, reducing attestation overhead to linear complexity under ideal conditions. It enables interoperability by enabling transitive trust across heterogeneous networks, and supports trust establishment with offline nodes via relayed attestations. Using a custom discrete-event simulator, we show that Careful Whisper propagates trust both faster and more widely than naive approaches across various network topologies. Our results demonstrate that our protocol is resource efficient, sending ~21.5 KiB and requiring 0.158 seconds per round in a 200-node network, and that our protocol is resilient to attestation failures across various network topologies.