The testing process is running as normal. The only things that changed with Operation Warp Speed are that no FDA bureaucrats can let the approval paperwork sit on their desk for six months while they take naps and drink on the job. And the other thing that's different is that President Trump negotiated with the manufacturers to have them begin making doses DURING the phase 3 clinical trials. That's a huge risk because it's extremely expensive to retool a factory to make your specific vaccine and there's no guarantee any of those doses will ever be approved. But doing that saves MONTHS of time to actually deliver doses versus waiting until after phase 3 trials are completed and approvals have been granted to even start retooling.
But the testing process remains the same as always. You run phase 1 clinical trials with a small group of closely monitored volunteers (~100 people). That looks just at safety, making sure you aren't going to kill anyone. Then you can move on to phase 2 clinical trials with a larger group of closely monitored volunteers (~1,000). Here you're mainly looking at what side effects are present and you get a little data about how well your candidate works. Then you can move on to phase 3 with tens of thousands (~30,000-50,000) closely monitored volunteers. This is where you get the full picture about both the safety and the efficacy of what you're testing. This must be completed before approvals are given.
They're all doing phase 1, phase 2, and phase 3 clinical trials for all these vaccine candidates. They have to demonstrate safety at every step of the way. If there's any question, the whole thing gets halted. The Oxford vaccine candidate has been held up for weeks because out of ~50,000 people, 1 person in the UK was in the hospital with a possible reaction to the vaccine candidate. After an investigation, UK medical authorities cleared the trials to continue, but the FDA has held them up for weeks over it. Anyone saying these are being rushed isn't paying attention.
That is just patently untrue. They completely skipped the animal trials, which is where there would regularly be a challenge test of the subjects subsequently being exposed to the virus. This is where similar vaccines have failed in the past, because they have intensified the disease and led to death. Also, they are also not waiting even the usual time yo see how the human subjects respond to the vaccines in this case.
The UK trial also had two similar and extreme reactions in their group, which is legit cause for concern.