If I understand you correctly, you need the Beanstalk application to SSH in to another server?
Every EC2 instance gets launched with a designated keypair. You have the option of either creating a new keypair or using a keypair already set (i.e. the keypair created by the Beanstalk application for the first instance).
Keeping the private key on the Beanstalk instance, launching the other instance(s) using that same keypair would allow the application to use the private key to SSH in and also allow you to scale the instances without your having to go in to each one and create new keypairs.
That said, I believe the documentation suggests against keeping the private key on the instance, so perhaps consider launching the non-Beanstalk instances with a configuration script that creates a customized user, perhaps using a key and password and pre-configuring the application with that information? You can keep that information as environment variables within Beanstalk itself, similar to how you would keep RDS credentials.
http://docs.aws.amazon.com/AWSEC2/latest/UserGuide/user-data.html
When you launch an instance in Amazon EC2, you have the option of
passing user data to the instance that can be used to perform common
automated configuration tasks and even run scripts after the instance
starts. You can pass two types of user data to Amazon EC2: shell
scripts and cloud-init directives. You can also pass this data into
the launch wizard as plain text, as a file (this is useful for
launching instances via the command line tools), or as base64-encoded
text (for API calls).
EDIT 1
In order to SSH from computer A to computer B, computer A needs to have the private key in the .ssh directory and computer B needs to have the public key appended to the authorized_keys file in the .ssh directory, so that's perhaps why you don't see either key in the Elastic Beanstalk EC2 instance.
Since you have the public key within the authorized_keys file, you can simply replicate it to the DigitalOcean instance (once it's on the server, do cat public_key >> authorized_hosts) and since you're able to SSH in to the Elastic Beanstalk instance, you can simply take the private key from your computer and put it in the .ssh directory of the Beanstalk instance. That way, now the DigitalOcean instance will have just the public key appended to the authorized_keys file and the beanstalk instance will now have both the private key and public key as authorized to login.
That said, this is probably the most insecure way of doing this...I would prefer you generate a new key and use that to be able to SSH from the Beanstalk instance to the DigitalOcean instance.
Note, this is not the same as creating a new IAM user, though you can use IAM to simply create new key pairs.
EDIT 2
I guess it will be difficult for an EC2 instance to automatically obtain the private key upon being automatically launched, so the way I see it, you have three options;
1) EC2 instances can be (auto)launched with a custom "user data" script, which I referenced above. In that script, you can include the actual private key data (pretty bad idea IMO) OR have it obtain the private key from somewhere (e.g. SCP with username/password into some machine and download it). Again, all pretty bad ideas.
2) Embed the private key within your Beanstalk application. Not knowing what language your application is written in, it's difficult to determine how bad / good of an idea this is. If it's in Java, private / sensitive keys get embedded all the time, so I don't see why this would be any different. This seems to me to be a fine idea, iff this is an application developed specifically for this use case and will never be used anywhere else. I'd hate to see a developer accidentally deploy this app somewhere else and now that private key is potentially compromised.
3) You could create an AMI of the EC2 instance with the key embedded in it. Then simply instruct the autoscaling to launch a new instance of that AMI and voila, you will have the key in the .ssh directory. I tend to like this idea the best as it uses AWS resources for what they're intended, and I would think makes the key a bit more 'secure' (outside of compromising the EC2 instance itself, it will be much more difficult for anyone to access the key). This wouldn't add any additional scalability over option #2 as you can scale / deploy a Beanstalk application just as much as you can an AMI image. That preference is up to you.
NOTE, this of course says nothing about scaling the DO machine, assuming that's even a requirement.
