In theory, my wife and I do a budget at the beginning of every month, and we check with each other if we are going to spend anything that was not planned on the budget.
In practice, we are a bit more lax than that. We usually don’t get around to planning our budget until the month has already started. In the meantime, we spend what we need to spend on essentials like groceries and gasoline. Once the budget is planned for the month, we don’t check with each other for anything that is a planned expense that we already agreed to on our budget. Even if some essential or important expense comes up that is not on the budget, we will generally go ahead and spend the money if it is less than $100, and then try to make adjustments on the budget after the fact, if possible. But for anything unexpected that is over $100, we try to check with each other before spending.
However, our budget is a little bit flexible, and my wife and I each have our own small “personal money” budget allocation, that we can spend on anything we want.
To answer your specific examples of something like an appliance or a computer, my wife or I would never make a large purchase like that without agreeing on it first. If possible, it would be planned in our budget from the beginning of the month. Or for a very big expense, we might save up for many months in advance. The exception would be an emergency. For example, if our very old refrigerator stopped working, and we found out it would cost hundreds of dollars to fix it, then we would probably buy a new refrigerator immediately, using our emergency savings, even if we weren’t able to fit it into our monthly budget. But that is a decision we would always make together.