Not necessarily, and it was due to immigration, imperialism, emigration etc that we even have such an exchange of ideas and influences to begin with. Henry VIII for example looked at the old Roman sanitation systems when instituting Sanitation in London. Many rich landowners also used Roman techniques in the early Middle Ages to convey running water and remove effluence from their Castles...there were also examples of public latrines in some parts of England, albeit rare. No doubt sanitation, particularly public sanitation reverted to basic, primitive techniques as Roman facilities fell into disrepair during the interregnum, but that is not the same as saying such would have happened without Roman influence or that Roman influence was not indicative in our culture.