Your parner with the Romanian passport can legally work in all EU countries since January 2014. If you can also be considered as a family member, then you will also have the same rights as her, and be able to work in the same country as she does.
The tricky part is whether you can be considered as a family member or not. If you'd be married then it's easy:
If you are working in another country - as an employee, self-employed or on a posting, your non-EU spouse, (grand) children or (grand)parents can stay there with you without having to meet any other conditions.
During their stay in your new country, your non-EU spouse, (grand)children or (grand)parents should be treated as nationals, notably as regards access to employment, pay and benefits facilitating access to work, enrolment in schools, etc.
If you are not yet married, then it's up to the local authorities to decide whether you can be considered as a family member, or not:
The authorities in the EU country where you are working will assess whether your non-EU relatives/partner should be authorised to stay there as your family members (because you are an EU national employed, posted or self-employed there for longer than 3 months). The assessment will be done on the basis of the national conditions and on a case-by-case basis.
You usually qualify if you can prove that you have been living together for at least two years.
If you qualify, then you will need a partner visa to enter other EU countries (especially if they are in a different zone, e.g. from Schengen area to the UK or Ireland)), but only in case you are travelling alone, without your partner. In other cases a "EU family member’s residence card" should be enough. The latter will also prove that you have the right to work in the country you are both living.
Working in a different country where your partner lives is a bit more trickier. In general to be considered for the same benefits as your partner, you both have to be a resident in the same country, so if you intend to live in a different place as your partner, then you are probably out of luck, especially if you want to do this for more than 6 months.
This shouldn't prohibit you to be a cross-border self-employed worker though (live in Germany, but work in the Netherlands for example), but I couldn't yet find any information about whether you'll have issues getting a job by being a non-EEU resident in another country (as you will only have a German residency card). Probably having your consultancy company registered in the same country as your partner lives would suffice though.