Different parishes do it in different ways.
In one parish in Fort Worth, for example, the parish knows to bring their objects up to the communion rail prior to Mass, and after communion/before announcements, the priest will take a few moments to bless the items. Then after Mass, people come to collect their objects that had been left up there during Mass.
In my parish, we share a priest with another parish, so I know his schedule is really tight to wrap up here and get on the road so he can meet his obligations at his main parish. So if I ever want something blessed, I know the best time to do it is after a weekday Mass, when his schedule doesn’t have anything pressing.
When I do want an item blessed in a specific way, I make sure to bring the prayer (along with context so he knows that the prayer is okay) along with the object. Like, for example, my blessed salt. I’ll bring my 1962 Missal, and have the page bookmarked, and say, “Please pray this prayer here over my salt.” They’ll usually look at the salt with a funny expression, look at the prayer, look at the book, thumb through the pages a bit, and then do the prayer. But not all items require specific blessings.