The reptilian species that are classified by our taxonomic system under the super order Dinosauria went extinct some 65 million years ago. No individual animal known to man has a lifespan more than an infinitesimal fraction of that. As such, we can presume the question doesn't refer to individual animals.
Many may be surprised to learn that if we apply the question to animal species, the answer is still none. All of the dinosaurs were terrestrial species, they lived upon the land. No species of animal living today, including the insects, shared the land masses with the dinosaurs of the Mesozoic Era. Not a single one. Every species of terrestrial animal living today had ancestors, pre-cursor species that lived alongside the dinosaurs, they could not be here else, but that is not the same thing.
All of today's mammal species, for example, are descended from small, burrowing proto-mammals that lived alongside the dinosaurs, but they evolved; they are not now the same species. Even the duck-billed platypus (Ornithorhynchus anatinus) of today, which is a monotreme rather than a marsupial, oldest of the of the taxonomic class Mammalia, did not live alongside the dinosaurs. There have been very few monotreme fossils found, but the earliest so far dates from the early Cretaceous period, some 112 to 121 million years ago, during the time of the dinosaurs. But it is not a fossil of today's platypus, it is of a pre-cursor platypus species called Teinolophos.
Just like individual animals, species have limited lifespans. They get superseded by their descendant species and die out; it is all part of how evolution works. Species lifespans vary, from as little as a few thousand years to many million, but very few last more than the 65 million years since dinosaurs went extinct. No species of land animal currently extant has survived that long. Some marine species probably have and may even have predated the dinosaurs, but whether they can be considered to have lived "alongside" the dinosaurs is dependent on how you define alongside.
Even the marine species still living today that may have lived at the same time as the dinosaurs are few. There are a number of jellyfish species that have possibly existed for a very long time. But even with them it is hard to say; having no bones, jellyfish fossils are rarely found. Jellyfish fossils deposited in sandstone some 500 million years ago were discovered in a Wisconsin quarry in January 2002. But it is difficult to determine from such remains if they were the same as the jellyfish species of today. Some fish species found now, such as the green sturgeon and the coelacanth, appear to be the same as fossilized remains of prehistoric species from the Mesozoic Era.
To be certain of "animals" of today that lived alongside the dinosaurs however, we need to go higher up in the hierarchical taxonomic classification system, to the level defined as families. At this level, we get far more marine animals, such as sharks and turtles, and start to get some terrestrial animals as well. Most of the amphibian families and many of the lizard and insect families of today had species living during the Mesozoic Era, and quite a few predated that time.
Individual insects have relatively short lives, so insect species tend to be quite short-lived as well; although they normally become extinct due to speciation, evolving into new species, rather than from dying out altogether. Species of dragonflies, for example,have lived on the Earth since the Permian Era, around 280 million years ago, therefore predating the dinosaurs. Although those of today are of the suborder Anisoptera, that has "only" been here since the Jurassic period some 180 million years ago.
In the near future we may be able to tell if anatomically similar fossil remains from prehistory are the same as currently living species, but we can't as yet. As such, we cannot say that any currently living species shared the world with the dinosaurs. We can, however, state that a number of taxonomic families of animals now living did indeed co-exist with those magnificent reptiles when they dominated the lands of Earth.