Is there a way to query these databases without transferring them to Google BigQuery project?
No. BigQuery operates against BigQuery projects and datasets.
Equivalently, there is a way for BigQuery to communicate with one of these datasets directly?
Equivalently, no. For exactly the same reason.
If it is not possible for BigQuery, then is there an equivalent service in Amazon EC2 that can do the same thing?
No, not really. There's Amazon Cloud Search, but it basically operates on the same principal and requires that you upload the data to be searched. So unless somone has already uploaded that data into an Amazon Cloud Search account, no, there's no eay to do it without uploading the data.