What are these dinosaur's competitors?
According to Wikipedia, a velociraptor is up to 2m long and 15 kg. A Deinonychus is up to 3.5m long and 75kg. I'm going to assume that those masses are the upper range for a large, healthy specimen, and compare to modern species accordingly. Deinonychus is in a weight class with only the Jaguar (100 kg) and Cougar (80 kg). Velociraptor has a wider range of competitors, including the Ocelot (15kg), Jaguarundi (10kg), Coyote (12 kg), Racoons (12 kg), Coati (12 kg) and Otters (15kg).
What are a dinosaur's characteristics relative to its competitors?
Family care and growth: Here mammals like the cats a have an unmistakable advantage. Milk allows them to provide a highly nutrient dense food source to their offspring, speeding growth. Mammalian growth rates are significantly higher than all other groups for body masses greater than 10 kg. Birds can exceed mammal growth up to about 5 kg; but growth rates for an ostrich are in the 200g per day range up to 20kg, which is about average for a mammal that size. Velociraptor might come out as a wash against similar sized mammals, but Deinonychus is almost certainly going to develop more slowly than a big cat cub. Advantage: Mammals
Intelligence: Velociraptor has come under extensive scrutiny, what with the opening of doors in Jurassic park and all. However, it appears that a raptor's ecephalization level is below that of cats and dogs, and its brain may have had a more primitive structure, with more space less efficiently allocated to sensory inputs. Also relevant is rate of intelligence growth. Milk's big advantage is its high fat content, which is critical for early, rapid brain development. Even if dinos get to be a smart as cats one day, a cat's brain has the raw materials to develop faster. Advantage: Mammals
Vision: A primary advantage of bipedalism is height of eye. The higher up you are, the more you can see. This is particularly important for seeing over tall grass. It might also be important in ability to chase prey into water. For the larger, taller Deinonychus, this might allow it to be a better hunter of capybara than the jaguar, for example Advantage: Dinos
Endurance: One proposed advantage of bipedalism is increased endurance. There is, for example, evidence that humans are better endurance runners than horses. This is hypothetical, but we'll score it Advantage: Dinos
Agility: Bipeds are fundamentally limited by rotational inertia. Therapod dinosaurs are further limited by a relatively rigid spine to help keep their body efficiently balanced on two legs. Cats, on the other hand, have very flexible spines. If a dino wants to change direction, it must swing its head and tail both around, generating torque around the axis of the two rear legs with the same legs that must be used for running. A cat, on the other hand can use its abdominal muscles to bend its entire body, so that its front legs start applying force in a different direction from its rear legs. Thus, a cat can generate torque from the same motion that it uses to run, while a therapod must use a different motion. Quadrupedal mammalian carnivores are more agile that bipedal therapod dinosaurs. Advantage: Mammals
Climbing: The cats in particular are good climbers, and the coatis are semi-arboreal. Despite being feather covered, Deinonychus is certainly too heavy to be able to get any lift; a jaguar's climbing ability is probably superior. Velociraptor might to better, but the manipulative hands of a coati will put it far ahead in a climbing competition. Advantage: Mammals
Swimming: Neither dino appears to be optimized for swimming. However, the otter is. It is highly unlikely that velociraptor could compete with an otter in the water. Advantage: Mammals
How does Deinonychus match up?
Not very well. First off, neither the Jaguar or Cougar is a cursorial (running and chasing) predator, instead they are ambush hunters. Deinonychus seems to have several disadvantages as an ambusher. First, as a poor climber it will have a hard time jumping out of trees. Second, being taller, it will likely have a harder time stalking its prey.
Third, there is an open question about how effective its 'attack' would be. I know they made a big deal out of the claws in Jurassic Park, but no animal that large today kills with a single disemboweling thrust. If a jaguar hunts a tapir, it has to tackle it and choke the life out of it. I don't see even a foot long sickle doing that much damage to a thick, muscled tapir hide. Jaguars can put crushing bites on animal skulls, with bite forces around 5000 N, while musculo-skeletal estimates of a Deinonychus are more like 1500 N (the 4000-8000 N estimates come from tooth indentations, and I consider them highly suspect).
The advantage of endurance could be significant in open terrain, but open areas for long chases are going to be limited in Costa Rica. This is an environment with lots of trees and tall grasses (taller than a Deinonychus, at least). In most parts of the country, a chase that lasts for multiple minutes will give the chased animal somewhere to run through for cover. Here, the superior agility of quadrupedal prey items will come into play.
Finally, and most damningly, there is no modern analogue for a bipedal big game hunter (unless you are using a bow and arrows). Ostriches and ratites are mostly herbivores or frugivores, and nothing else is that close in size. Deinonychus won't make it.
How does Velociraptor match up?
Velociraptor's disadvantage is that it is competing with a wide variety of mammals specialized for different habitats. It is tough to compete with otters in the water, coati in the trees, and ocelots and coyotes on the ground. Fortunately, since we can assume velociraptor will survive on much smaller prey, it has a lot more options for niches.
The upright vision aspect gives velociraptor a significant advantage in hunting small animals in open clearings. It will be able to spot prey from farther away, and track it better while running. Its smaller size will minimized the negative effects of rotational inertia on its ability to turn. In addition, it may be able to use its feathers to perform a leaping/gliding attack to surprise small burrowing prey (like gophers) from above.
The best point in its favor is that there exists an analogue on Earth, in a niche currently unoccupied in Costa Rica. The secretary bird is a mostly terrestrial, avian, dedicated carnivore. That is, unlike most other birds in its niche, it is not a primary insectivore. It eats a lot of lizards, rodents, snakes, tortoises, small birds, and even small predators like mongoose. The secretary bird is capable of full flight, so its mass is limited to 5 kg, but a larger, non-flying, more-Velociraptor-ish variant is feasible. The larger velociraptor would be able to expand the secretary bird's dietary range by tackling the young of larger prey mammals (peccary piglets, deer fawns, etc) as well as the variety of medium-small animals like rabbits, opossum, paca and agouti.
The velociraptor's closest competitor in this niche would be the raccoon, but the raccoon is nocturnal and tends to stay in areas of heavy cover, since it depends on hiding to escape predation. The velociraptor, on the other hand, would operate during the day in open clearings, and use speed and endurance to escape things big enough to kill it, like a jaguar.
Velociraptor seem like they have an available niche, so they might be here to stay. They would probably be strongly selected at first in the direction of the secretary bird: longer legs, more relative wing power for gliding attacks, and keen daytime vision (if they don't have that already).