How do I separate separate iPhones that are using the same iCloud account?

Log out of the iCloud account on one of the phones and create a new Apple ID for it.

Apple IDs are personal; they shouldn’t be shared any more than you would share a toothbrush. If you share an Apple ID with someone else many bad things will happen. Each user should have their own Apple ID, then you can create Family Sharing→What is Family Sharing? - Apple Support. That way you can share what you want to share and not share what you don’t want to share.

It is OK to share an Apple ID among devices used by one person, such as an iPhone, an iPad and a Mac; in fact, it is a very useful thing to do. But not between devices used by different people.

Here is Apple’s FAQ: https://appleid.apple.com/faq/#!&page=faq. See the 4th bullet item.

And here is Apple's expansion of what I've posted→Make sure that each family member has a unique Apple ID - Apple Support

Using the same Apple ID on two devices is meant to integrate them, so you could start work on one device and continue on the other or maintain the same contact list. Having another Apple ID may be the best solution for you depending on what you want to separate.

If you are just looking to receive messages on a certain device, then that can be accomplished by going to Settings > Messages > Send & Receive, and changing both the send section and receive section to just your phone number and not your Apple ID. Now the messages will depend on which phone number the message is sent to and will not show up on both devices. This would need to be done on both phones.

Specify what services you would want to act independently and I may be able to provide a better answer. As mentioned by Lawrence Finch, a unique Apple ID should be used for each person. Not clear if you are using both of these devices for personal use or used by two different people.