Are there pockets of liquid carbon dioxide in Earth's oceans?
Yes, according to this paper in the Proceedings of the National Academy of Sciences. But they also say that this "lake" of liquid CO2 is covered by CO2 clathrate, so it's not quite your average brine pool.

Note about clathrates:

The following image is from the Wikipedia article on CO2 clathrate. It is a CO2 hydrate P-T phase diagram with experimental data (black squares) from a 1998 paper by Sloan listed in the references of the Wikipedia article.

Note the H2O phase lines drawn in. There is a relatively small habitable area for clathrate somewhere inside about 273-280 K (0-7 ºC) and 10-12 bar (or about 90-110m in ocean depth as calculated here). Above the phase boundary, the clathrate will dissolve to liquid. Below, to gas. What this diagram does not say (according to a knowledgeable colleague) is that clathrate will not readily form or remain in its semi-organized state given very slight amounts of mechanical agitation. Hence why it does not exist en masse under typical seafloor conditions.