But it is quite common for someone to take the same problem, collect their own data and run their own analysis (both steps can be completely different from the original statement-- collecting data of a different type, from a different instrument, etc. Running an analysis of a different paradigm, running the same analysis to a different precision, running the same analysis via a different algorithm, etc.).
If you read the arxiv on a daily basis you can see huge academic arguments unfolding over the course of months and years.
There seems to be this idea that the conversation goes: "Yo I found this hypervelocity star" "Dope, let's move on"
It's actually more like: "Yo I found this hypervelocity star" "Nope, I got spectra and you're wrong" "Well I got ultra high-res spectra and I think he's right" "Actually all of you are forgetting asymmetric drift, this is just a geometry problem, l2angles" "Hey, I sit in my basement and play with MOND, it might help" "My simulations show something completely different"
Authors are called out and proven right or wrong on a daily basis, even if we can't watch them code over the shoulder. I actually think that's one of the beauties of it-- most of our methods are invented by trying to prove or disprove something in a new way.
The monster codes like GADGET (which ran the millenium sim everyone's seen) are usually made public after ~5 years of proprietariness.