The attacker already holds working cloud credentials, either stolen long-term access keys, a compromised IAM role, or a hijacked SSO session, and uses them to call a sharing API instead of a download API. On EC2 that is ModifySnapshotAttribute with attributeType=CREATE_VOLUME_PERMISSION and operationType=add, which grants a chosen AWS account the right to create a new volume from your snapshot. RDS has the same primitive through ModifyDBSnapshotAttribute and ModifyDBClusterSnapshotAttribute. None of these calls move a single byte at the moment they run; they only change a permission.
For S3, the equivalent is PutBucketReplication, which configures a bucket to continuously copy its objects into a destination bucket that can sit in a completely different account. Once the rule is enabled, replication runs automatically and keeps running until someone disables it, which is what makes it attractive for sustained exfiltration rather than a one-time pull. A narrower variant uses PutBucketPolicy to add a foreign account as a Principal with s3:GetObject rights, or generates a pre-signed URL that lets anyone with the link retrieve an object without any AWS credentials at all.
The actual data movement happens later, and often from the destination account, which is why sharing and transferring look like two separate events in the logs. The attacker (or an automated pipeline they control) calls CopySnapshot or launches a volume from the shared snapshot, or CopyObject/GetObject against the replicated bucket, entirely inside their own account. If you only monitor the source account you frequently see the permission grant and nothing else, because the actual copy never generates a log event you have access to.