There was some dalliance with duplicating a SIM or account with multiple phones but I would be guessing you wanted to have one account across more than one phone. In that case, yes. I would expect texts to arrive on both phones.
You didn't tell enough but in 99.99% or higher cases, the answer is no.
Bob
If I give my child my old cell phone to use to play games, will any text messages I send on my new phone show up on my old one?

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