Who is supposed to repeat experiments?

The gold standard is blind, independent replication. However, this almost never happens because it costs a similar amount (of materials and time) and there is less benefit to the researcher who repeats the experiments, so there is little incentive. Next down would be independent replication by another group, then replication within the same group. The bare minimum is that the experiment be described in sufficient detail to be reproduced. For example, the 'Instructions to Authors' in the Journal of Infection and Immunity (which is fairly typical) states:

the Materials and Methods section should include sufficient technical information to allow the experiments to be repeated.

Reproducibility is one of the most important issues when describing or reviewing a scientific paper (ref). Knowing that a result is less likely to be attributable to a chance combination of uncontrolled parameters, or a mistake on behalf of the scientist conducting the experiment or recording the experimental conditions, increases confidence that the result seen is a 'genuine' phenomenon. As Karl Popper, the founder of the scientific method, put it:

Non-reproducible single occurrences are of no significance to science.

(Popper 1959, The logic of scientific discovery. Hutchinson, London, United Kingdom.)

Unfortunately, most journals place a high premium on novelty and therefore it is more difficult to publish reproductions of previous studies, and when these are published they are often in less important journals. Funders, similarly, direct reviewers to score grant applications based, amongst other things, on novelty. This greatly reduces the incentive to replicate published studies - if you can't get funding to do something and you wouldn't be able to publish it anyway, it won't get done. Two other major and common obstacles to independent validation are:

1. that it requires the involvement of another lab with the same resources and skills, which might be very rare;
2. For experiments involving animals specifically, there is an ethical requirement to minimise the use of experimental animals.

The lack of reproducibility of experiments is a problem in most areas of science and awareness of this is growing. In 2012 this prompted the Reproducibility Initiative, where researchers can pay a fee for blind, independent replication prior to publication.

This sounded like a step in the right direction at the time but as far as I know awareness of the initiative remains fairly low, uptake has not been great and 'proper' reproduction remains very rare indeed.

I'm not sure anyone is supposed to repeat experiments, in any formalized way. In engineering, and I suppose physics, you end up repeating experiments because you are looking to use a method to solve a problem. So if someone has published a method to solve a particular problem, researchers will often try it in order to see how well it works for their system. Some methods work better than others, and I guess the best performing methods will be cited and used and become well known. Methods that don't work, or method with flaws and errors will tend to be forgotten over time.

Those whose research benefits from it

While it would be nice to repeat experiments, in practice we as a society don't really want¹ scientists to repeat them just because, we want them to create new knowledge instead.

So experiments usually are repeated in three cases:

1. When a researcher doubts the original experiment - if they distrust the outcome or the procedure, then replicating the experiment (possibly with modifications!) is useful science, since it creates new knowledge in addition to the initial result, no matter if the replication yields similar or different results. A negative result would definitely be publishable, and a positive result would be publishable iff the wider community also consider the first experiment as shaky and needing confirmation.
2. When a researcher proposes an improved method or theory, they will often reproduce the previous experiment to make a proper comparison instead of just comparing reported numbers.
3. Some experiments are replicated during teaching process, either to illustrate a concept to students, or to teach grad students state-of-art techniques before setting them on new experiments.

However, for experiments that are both hard to perform and seem trustworthy, there is no incentive to repeat them, it would not be particularly useful for the required effort. If you're really sure about the results that you'll get, then you don't gain any information by doing it, and that's not research anymore.

1_{Ignoring empty words, this is clearly illustrated by actions of funding bodies and university leadership who are setting the direction where scientists are applied.}