DH and I have always had separate accounts - we have a joint account in 1 institution which we HAD to open to allow mortgage cheques go through when we were doing an extension and needed multiple drawdowns from the account, we now use that as a spare savings account.
He has always paid the mortgage, I have generally paid the utilities, insurances and food bills, we each pay for our own cars (although we often will cover something for the other as needed) etc.
We each have our own savings accounts and provide for our pensions (because we have slightly different habits - although we DO both have savings).
I tend to do most of the buying of DD's clothes, gifts for Christmas and birthdays, stationary supplies etc - but DH will organise the sports gear, school books, etc. And we split holiday expenses, often it depends mostly on who has time to organise all these things (I tend to build a stock of items in a gift closet situation, while DH is somewhat more last minute, but great at Amazon on the fly!).
If either of us is ever stuck, we will ask for help from the other. Mostly it's offered before it's needed - even things like knowing that I keep a small stash of cash in the safe (mostly so I can always find money for a takeaway delivery on a Friday night if I really need to!), and just letting me know to top it up, or that one of us has a lot of travel for work so needs their credit card available for hotels and can the other cover the general household for that month etc.
For us, it works. But we are open with each other about where things are and our needs. We earn enough that we have flexibility. And we look at the overall budget together periodically to see if the balance is fair between us (not necessarily 50/50, but suitably proportional so we each have spare capacity and looking at what things its easiest for each of us to pay).