All databases of which the flows are mapped in accordance with openLCA standards can be imported directly into openLCA regardless of whether they have been downloaded from openLCA Nexus or not. HOWEVER, this is often only the case if a database is from openLCA Nexus. ... so it is about flow mapping and not about openLCA nexus.
If you want to use a database in openLCA the database's flows must be mapped in accordance with openLCA standards. This is the case for all databases available in openLCA Nexus. However, this is NOT the case if you download ecoinvent directly from the ecoinvent website. Thus an ecoinvent database downloaded from the ecoinvent website cannot be direclty imported into openLCA. You should download ecoinvent from the openLCA website.
To my knowledge there is no such thing as an 'ecoinvent free' database. However, academic/educational users from non-OECD countries receive ecoinvent for free.