There are going to be many ways to build micro-services. I would start by familiarizing yourself with the whitepaper AWS published: Microservices on AWS, Whitepaper - PDF version.
In your question you stated: "The issue now is communication between services are via API calls through the API gateway. This feels inefficient and less secure than it can be. Is there a way to make my microservices talk to each other in a more performant and secure manner?"
Yes - In fact, the AWS Whitepaper, and API Gateway FAQ reference the API Gateway as a "front door" to your application. The intent of API Gateway is to be used for external services communicating to your AWS services.. not AWS services communicating with each other.
There are several ways AWS resources can communicate with each other to call micro-services. A few are outlined in the whitepaper, and this is another resource I have used: Better Together: Amazon ECS and AWS Lambda. The services you use will be based on the requirements you have.
By breaking monolithic applications into small microservices, the communication overhead increases because microservices have to talk to each other. In many implementations, REST over HTTP is used as a communication protocol. It is a light-weight protocol, but high volumes can cause issues. In some cases, it might make sense to think about consolidating services that send a lot of messages back and forth. If you find yourself in a situation where you consolidate more and more of your services just to reduce chattiness, you should review your problem domains and your domain model.
To my understanding, the root of your problem is routing of requests to micro-services. To maintain the "Characteristics of Microservices" you should choose a single solution to manage routing.
API Gateway
You mentioned using API Gateway as a routing solution. API Gateway can be used for routing... however, if you choose to use API Gateway for routing, you should define your routes explicitly in one level. Why?
- Using {proxy+} increases attack surface because it requires routing to be properly handled in another micro-service.
- One of the advantages of defining routes in API Gateway is that your API is self documenting. If you have multiple API gateways it will become colluded.
The downside of this is that it will take time, and you may have to change existing API's that have already been defined. But, you may already be making changes to existing code base to follow micro-services best practices.
Lambda or other compute resource
Despite the reasons listed above to use API Gateway for routing, if configured properly another resource can properly handle routing. You can have API Gateway proxy to a Lambda function that has all micro-service routes defined or another resource within your VPC with routes defined.
Result
What you do depends on your requirements and time. If you already have an API defined somewhere and simply want API Gateway to be used to throttle, monitor, secure, and log requests, then you will have API Gateway as a proxy. If you want to fully benefit from API Gateway, explicitly define each route within it. Both approaches can follow micro-service best practices, however, it is my opinion that defining each public API in API Gateway is the best way to align with micro-service architecture. The other answers also do a great job explaining the trade-offs with each approach.