A European educated engineer here (Aerospace BEng), now working in the US.
My studies in Ireland included a 9 month work placement in industry with opportunities to work across Europe, many traveled to UK, France and Germany for placements. Here in the US there are similar internships through universities, although like Ireland, not all universities choose to offer this. I 100% recommend choosing a university which offers work placement as many students graduate with job offers from their placement companies.
As has been mentioned, Mech engineering covers many different industries and specializations. I studied aerospace engineering so I gravitated to that field and over the past 10 years have spent time in R&D, Materials, Manufacturing and Design roles within Aerospace. If you have graduated, you have likely specialized during your years at university already, based on electives, projects etc...fluid dynamics or materials science were two of the bigger paths in my own degree.
I have split my career to date equally between Europe and USA. The earnings/savings potential in the US is certainly greater and cost of living is lower but it is not an easy task to obtain a visa to work here (and may only get harder with the current administrations proposed reform of the H1B visa program). I have found the work-life balance better in Europe but that is not to say companies that value that culture don't exist here in the US, I know they do, but many have an almost mandatory overtime and live to work culture.
If you are looking to immigrate here, there are probably two options outside of marriage to a US Citizen.....H1B and L1 visas, and of the two, L1 may be the easier to obtain for the Mech Eng industry. This is the inter-company transfer of an employee. You work for a European company with USA locations or US company in Europe and then transfer to the companies USA location when you are eligible and there is an opportunity to do so. Automotive is probably the obvious choice as mentioned but there are many more. Aerospace would be another, my company has european locations, we work regularly with Airbus Hamburg who have a number of USA locations and there are many German suppliers to US aerospace companies with locations in the USA, one example is Diehl.