That's an interesting question. I personally think they had their own names. I mean, I can see them addressing each other as "Venus" or "Mars", etc, or going by "Princess of [insert planet here]".
It's like some Power Ranger stories I've read. In some of those, the writer made it so the rangers refer to each other by color (like, say "Pink, we need backup here!") when they are within earshot of normal people. And that would make sense, as a way to further keep their secret identities, even if their faces are hidden.
So I can imagine the Senshi calling each other that the same way, or formally going by those names. But personally, I think that they would have had different legal names. Most stories just give them their modern ones, and that works. And I know there was a fic that made it so their past life names were variations of their modern ones.
I remember the story, but it's rather long so I can't look for the example there, bu it was similar to what they did in the English dub of the anime, how Princess Serenity was reborn as Serena. See what I mean?
And personally, I would think that if their actual names were the same as their planets, that would lead to a lot of confusion. Like, if Mina's name was just "Venus", then in a conversation, you might not know which "Venus" is being talked about, the planet or the person.