The author has the final say, but if you send a manuscript to two markets, and they both accept it, you will never sell to either market again.
What you are describing are simultaneous submissions. They are almost universally frowned upon. You should never send out simultaneous submissions unless you send them solely to markets who specifically state in their guidelines that they take simultaneous submissions.
The proper procedure is to send the manuscripts out to one market at a time. While you are waiting, write a manuscript to send to a second market.
If you get caught sending simultaneous submissions and both sell, you are creating major problems. If you withdraw the manuscript, it upsets planning. There have even been times when authors withdrew a manuscript after the magazine had set it into type and scheduled it for publications. So the editor has to remake the issue, and the typesetting time was all wasted. That editor will not trust you again, so you can write that market off. The editor will also talk to the editor who published your manuscript, who will stop reading your work for fear you will pull the same trick.
There is a difference for nonfiction, since you’re usually sending a query. AFAIK, it’s generally acceptable to send multiple queries, but the article can only be used exclusively. If you get two requests, send the article to the first on you received; if they arrive the same day, send it to the better paying market. Since the editor is not spending time and money (he can still reject the finished article), there’s no problem if you say “no.” Once you send the requested article, though, the same rules apply as for fiction.
Some argue that no simultanous submissions slows the process. However, it actually keeps the process moving faster. If there are a thousand writers sending to ten magazines, each of those magazines would get 100 manuscripts. If, however, the writers were allowed to do simulatenous submissions, they will all send to the ten magazines at once, so each magazine would get 1000 manuscripts. This would slow response time to ten times what it is, since the editors would have to read ten time the number of manuscripts.