My boys both had strong personalities when they were babies...which carried into the present. They are both strong-minded kids!

I say they are born with it

Well, as a Registered Midwife, I've helped bring 164 of those future ankle-biters into the world and I'd say that they have the seeds of their own personality already growing within them.
For example, when my daughter was born the doctor was astounded that her middle finger on her right was firmly raised and the rest of her hand was clenched tightly, almost as if she giving the world 'the finger.'

Research and developement say it takes about 4 months for hormones to settle, birth trauma, and everyone to just get into a pattern.
So while it starts instantly you mostly start to see a child’s personality developed around the 4 month mark, however this is just the start.

I would think we are born with a certain temperament and nurture/environment helps mold that into a personality.

i think its something the mother will pick up on first because of that connection. its nature and nurture

Depends what you are calling and including in personality. Babies are born with certain innate instincts and may have a particular way that they respond to stimuli that trigger these instincts.

Most of personality is learned after birth. It is taught by the reinforcement of behavior their parents and others in their sphere of influence. I have not studied the debate between Nature vs. Nurture in a long time. Do not know where the debate stands nowadays. But the first three years are incredibly important in the formation of the personality. Sorry for the boring quasi-academic answer.

From birth, my two kids were different in their personalities. I think that you are right that they are born with a personality. You see it more as they develop.

It is amazing how this happens. They are born as little bundle of living flesh and from nowhere comes a person. In my experience (3) about 6 weeks or less. Of course it becomes more pronounced as they grow but I slightly envy you your discoveries.

they are their own little person at birth then elaborate on that in their early years. its nature and nurture both