Some enzyme-catalyzed reactions always go in the same direction in a live cell (e.g. reactions of citric acid cycle, RNA polymerase) and others go in either direction depending on circumstances (for example, some enzymes of glycolysis are also involved in the reverse pathway that makes glucose).
As for reasons why a reaction might go in one direction only, you are on the right track. You mentioned two reasons you could imaging for a preference in one direction:
Does this mean that rate constants are very different in value depending on direction?
If a reaction is far from equilibrium (i.e. proceeds mainly in one direction), the forward rate and the reverse rate will be very different from each other. This could be because the rate constants a very different from each other (with similar concentrations of reactants and products) or because the concentrations of reactants and products are very different from each other, or both. So the safe thing to say is that the rates differ by a lot (rather than the rate constants).
Or the reason is that reaction product molecules are always quickly consumed by the next step, and as a result the concentration of the product is always low?
This case also occurs. In RNA synthesis, for example, the reaction of a NTP (i.e. a triphosphate) with the growing RNA occurs near equilibrium, yielding longer RNA and diphosphate (pyrophosphate). In a second step, pyrophosphate hydrolyzes quickly, removing one of the products and making the first step virtually irreversible once the second step has occurred.
In the end, whether the reaction goes in both directions or just in one is not dependent on the enzyme: the enzyme is not capable of changing the position of an equilibrium. Whether the reaction goes forward or backward depends on the equilibrium constant of the net reaction (which the enzyme is not a part of) and the concentrations of the reactants and products.
You could envision a scenario where all enzyme-catalyzed reactions happen to be far from equilibrium under physiological conditions, but this is not the case, as the counterexamples mentioned above show.
Enzymes that catalyze reactions that are far from equilibrium are often regulated (turned on and off in various ways) to regulate entire pathways. If a reaction is near equilibrium and is used in different directions by distinct pathways, regulating the enzyme catalyzing the reaction would affect both pathways, so usually these enzymes are not regulated.