By default splunkweb runs on port 8000 so it should be http://my.private.ip.address:8000. If they have done something different, I am sure the dox say so; have you read whatever is provided in AWS marketplace for this (it is the very first question everyone would ask so I am sure there is some kind of FAQ answer for it in the notes)?
If you want a public IP assigned to your Splunk Enterprise running on AWS, you will need to launch the EC2 instance in a public subnet either in your default VPC or a user-created VPC. Also, if you use a user-created VPC, the public subnet will need to be enabled to assign public IP addresses to newly launched instances since that flag is false by default. Here is additional information from AWS on IP addressing of your VPC. http://docs.aws.amazon.com/AmazonVPC/latest/UserGuide/vpc-ip-addressing.html