Well, this goes along the line of someone with tulpas, or basically sentient personalities that are all individual. The text might come across as third-person if written in this way.
I've spent some time with people with said personalities, and their way of differentiating each personality is by saying each line of text differently, sometimes with a different pronoun (as not all personalities have to be the same gender.)
Besides making each name be at the beginning of the dialogue, depending on how the story is written, and whether the person is communicating by text or actually speaking can all effect how clear it is to the reader, because the tone could vary. In general, it might be best to avoid making personality changes too often, unless you are intentionally trying to disorient the reader. But, let me show a little bit of a demonstration.
Avoiding Gender Pronouns:
- Michael was cold, but so was Jake, the temperature letting there be no peace no rest between the two in Michael's mind.
- Amber could never find the right shoes, but another silent opinion said otherwise.
Using Gender Pronouns:
- Jake loved the smell of fresh-baked cookies in his house, but she also thought it never smelled fresh. It seemed like nothing was perfect for her, even if it was for himself.
- Michael took a good look at himself in the mirror, but who was he really? One one hand, there was Amber, and she was nice at least. On another hand, there was Jason, and hearing that voice tell him how awful he looks today just makes Michael want to snap.
In Dialogue, Including Another Character:
- Jon looks to his friend in front of him and asks politely, "Why is it you seem so down today?" There is a small silence between the two. "Oh, it's just about what happened last night." This time, Jack decides to reply openly, "Well, you need to get over it already."
- His friend was lingering on the response from Michael, thinking how rude this is. "Why did you have to say it that way?" However, this time, Kerry comes to the front of Michael's tongue, speaking in a different tone. "Oh, I didn't mean that at all."
Experiment with yourself a little, see which way you like the most.
I hope by some means this was a helpful insight, I wish you luck with your writing!