Curious, could I ask the country? I've lived in both the US & UK and would say that the vast majority of companies I've been involved with (via employment or simply interviewing) haven't cared about my degrees, but instead care about what I've worked on in the past and what they perceive I could learn/add to the company.
NB I don't have a CS or EE background.
Portugal.
Without a degree very few companies will bother.
In France, Germany and Switzerland one should at very least have a technical experience level and written references from past jobs.
It's good we (still) have free-ish movement of labour within the EU. (And closer to the ideal of free in IT than in almost any other sector.)
Well, kind of.
There is also the language barrier.
Not many accept working in English, and we had some customers from well known multinationals that only upper management was willing to use it.
Besides the degree, knowing multiple languages and soft skills really helps.
On my own, if you don't come from CS or EE background you hardly will get through HR.