So it was definitely intended to be secret — the reason is because they wanted to surprise Japan with the weapon. This presented obvious problems with keeping it secret when they were exploding a massive bomb.

Their main way to keep it secret was to do the test in relative isolation. The Trinity Site was in an area of very low population density near Alamogordo, New Mexico. It was not, however, zero population density.

Part of their strategy for keeping it quiet depended on how big the explosion was, which they did not, at that time, know. They could guess the theoretical maximum size and the most probable size, but they didn't really know what it would be. (The actual yield was 5X larger than their guess as to the most likely yield.) They also knew that detonating a nuclear weapon on the ground would involve a cloud of radioactive fallout spreading with the wind, and they hoped it would not go to inhabited areas, but they couldn't really control that, other than waiting for the weather to be good and hoping it wouldn't change.

So they spread military intelligence agents all around the test site, in the local towns. These agents kept a watch on the people to see if anyone talked about it. They were also in place in case evacuation proved necessary. They also stationed scientists nearby, with Geiger counters, to help track the spread of the radioactive cloud, and to raise an alert if the radiation levels got too high.

They also prepared a series of press releases to be sent out after the test. You can read them here. They range from the most understated disinformation (i.e., a remote ammunition dump exploded, but nobody got hurt) to catastrophe plans that involve revealing a lot (i.e. we were making a new kind of bomb and it blew up wrong and required evacuation of lots of people and killed a lot of top scientists).

As it happened, the test went fine. Nobody died, nobody had to be evacuated, the cloud more or less did what it was hoped it would do (except for some cattle ranchers' cows who got a bit overly irradiated — the government bought the cows and destroyed them). The "ammo dump" story was given to the local newspapers, and the military intelligence people monitored talk around the towns (e.g. in the bars, shops) to see if anyone was speculating on anything else to the contrary. People accepted the ammo dump story, and that was that, until after the Hiroshima bombing, when it was revealed was the truth actually was.

You might be interested in a previous question on a nuclear secrecy:

• Why was the design of Nuclear Submarines not given the same level of secrecy as the development of the nuclear bomb? - 2 comments, over 1 year old.
• One of the commenters is a flaired expert on Nuclear Technology and gives a summary of the type of secrecy employed in the Manhattan Project in general.

That same user, /u/restricteddata, might have something more specific to say about nuclear secrecy between the Trinity test and the atomic bombings, so please don't take this as the definitive answers.