If the server is truly "dual corded," as the term was meant to imply, then the answer is "yes." In fact, if the server is not drawing power from both supplies, in virtually equal amounts, something is wrong.
Believe it or not, there have actually been some "off-brand" devices put on the market with two power cords that are simply tied together inside the box and run to a single power supply. This allowed them to advertise "dual corded," even though the arrangement was not only misleading and useless, but also potentially dangerous. The intent in "dual corded" equipment is that each power supply is drawing power from a different source, and running at less than 50% of its capacity, with both supplies sharing the load equally at all times. In this way, if power is shut down to one cord, or if one internal power supply fails, the total load is instantly picked up by the other supply, which still has the capacity to handle the total load.
The same principle applies to your PDU's and UPS's. Nothing in the power chain should normally be operating at more than 50% of its rated capacity so it can pick up the total load if its "twin" suddenly fails, or must be turned off for maintenance. The difference is that you are responsible for keeping UPS and PDU loads in balance, whereas the server manufacturer makes sure the power supplies are designed and rated to accomplish this.
It should be noted that Power over Ethernet (PoE) has become something to be cautious of in this regard. The power supplies in a big network switch are also load-sharing, but they weren't necessarily designed to handle the load of a fully populated switch with PoE devices on every port, or perhaps not even on a significant number of ports. You can keep plugging in PoE devices, and the two power supplies will continue to support them and to load share, but they may very well each be running at more then 50% of capacity. In this case, if power fails to one side, or if a power supply blows, the remaining supply will be trying to run at more than 100% capacity, which will, of course, cause it to fail as well, taking the entire switch down with it.
When running PoE, it is wise to be certain how switch power is provided, and how much capacity is really available for PoE before power supply redundancy is compromised. This is not something the manufacturers make very clear when selling the PoE option on a switch.