Most 3D printed offerings these days are from two types of printer, filament printers or resin printers. The symptoms that have been mentioned above do not arise from one of these types of printer rather the another but from different ways of curing the plastic print.
Filament printers have evolved from the blobby light-but-strong Shapeways product which melted a pre-made nylon into a droplet which was touched onto the previous droplet that was still warm and the new hotter droplet fused with the older drop where they touched. The model cooled and the plastic set to build the structure but there was nearly as much space between the spheres as there was plastic within the spheres and the spaces soaked up vast quantities of paint which took years to dry if it was air-drying and a long time to cure if it was water-based paint. Filament printers today use the same principle but have a continuous feed of "stretched sprue" laid on a previous length so the whole proces is faster and has only two-dimensional voids between the threads. Paint stays on the surface of the threads and helps to smooth the surface steps from the printing process too. The model is built up from a machined steel base plate and the waterline is dead flat and stays that way. What is more, the model can be supported from the inside and requires no external support so these printers produce an ideal product!
Resin printers have to be built up on external supports, they use a softer plastic which flows over the layer below to produce a surface which is smoother than the steps on filament prints but the resin tends to sag between the external supports. The soft structure is then cured and that process often used UV light which cures the outside but can only cure the inside of the model if the plastic is transparent to UV light and that is why you are finding liquid leaking out when the outer surface is opened up. As Eric says, you can buy a UV light to cure the parts that had not previously seen the light. Most modern set-ups use microwaves to cure the soft plastic so the heat to cure the plastic is generated from the inside of the model and the outside remains cool. Microvave-cured models have a small amount of uncured resin on the outside and this is removed using a bath of water with additives (often these days a washing up detergent but car screenwash also works).
The problems with Shapeways FUD prints weeping oily liquids preventing paints from drying or curing has been solved and eliminated by changing the washing process after printing so you will not see that problem on newly printed models today from Shapeways agents.