My answer considers EASA (European) rules.
As a general rule after a transition training in simulator, there is line-flight training in real aircraft. That is flying with (experienced) instructor, although there are passengers onboard. The length of this line-flight training depends on previous experience of the trainee, but it cannot be omitted. After line-flight training there is line check (usually again on actual normal commercial flight) after which the trainee can commence operations without instructor pilot onboard.
In Europe, there is a concept of "in-experienced" crew member. After line-flight training and successfully passed line check, the pilot is considered in-experienced for at least 100 flight hours and 10 sectors. The key idea here is that two crew in-experienced crew members cannot fly together.
So, before two pilots are allowed to operate together, both shall have at least around 120-150 hours of flying the type under their belts.
There are some deviations on this, and most notably airlines can temporarily ignore the in-experienced status when introducing new aircraft types into their fleet. But what they cannot do is skip mandatory line-flight training. It is not uncommon for airlines to buy training for their instructors from other airlines which already operate the type or have instructors from other airlines flying with their newly acquired planes with company pilots.
The case of A350 is little bit different, as it is considered as a "same type" as A330 and they have common type rating endorsement. Many of new A350 pilots are already A330 pilots they can skip some of the line-flight training involved. There is some sim training in between.
With my operator I had couple of flights with instructor when transitioning from A330 to A350. There was no separate line-check or in-experienced status, as I already was active with A330. So I guess in theory it might be that two A330 pilots can fly A350 together after transition course in simulator.
Partial source: https://www.easa.europa.eu/sites/default/files/dfu/Consolidated%20unofficial%20AMC%26GM_Annex%20III%20Part-ORO_0.pdf