Do I need my own attorney during mortgage refinancing?

I've never used an attorney for a mortgage refinance before, and never wished I had either. There really isn't much more to a refinance of a mortgage than any other type of refinance (e.g. car, student, credit cards, etc). One bank gives you money (probably at a lower interest rate), and you pay off the other loan with that money. Typically there is a title company that handles all the checks and makes sure everything is up to snuff. I'm sure you could pay an attorney to look over the documents, and even represent you at closing if you wish, but I've never found it necessary. As long as you confirm the interest rate, the term, and the fees beforehand, there isn't much else that would require legal representation. One scenario where it makes sense to have an attorney is when you won't be present and you want them to sign on your behalf (called power of attorney).

Note the original purchase of real estate is typically far more complicated, and I would definitely encourage having an attorney in that case.

No, you don't need a lawyer. I've done several purchases & refinances over the years, and have never used a lawyer. Indeed, I've never even heard of anyone using a lawyer for a normal residential real estate transaction.

Note per comments: this applies to 49 US states, not Louisiana, where the system might well be different (I have no personal knowledge), and not to other countries.

The only time I could think it might be vaguely worthwhile hiring an attorney would be if your current lender has a significant Early Repayment Charge on your current mortgage, or if you are unsure how much your current bank will charge you to leave.

Often there isn't any charge at all if you've held the mortgage for a while and are outside of any initial fixed rate period, but if you're still inside the Early Repayment Charge period and aren't sure how much it's going to cost you, make sure to find out first.

If your bank is charging unreasonable interest rates and you're trapped by a similarly unreasonable Early Repayment Charge, then you may need an attorney.

But in all other cases as long as you understand the numbers - including any fixed costs of moving from one provider to another, I see no reason you'd need an attorney.