It really depends on what the task is and the concierge likes to do.
I had a reservation that I wanted to cancel with certain limitations. The concierge was not willing to bend the rules. So I got it done over the phone with WoH supervisor.
I also had a reservation cancelled on me. I sent emails to the concierge and asked her to rebook it. No response over two days. So I had to call Hyatt customer care support and they booked me an alternate reservation.