The article about highway=footway says it implies foot=designated in its description box. foot=designated can be determined by signage or by law directly (mentioning or drawing a specific way on an official map) or indirectly (eg. a law stating that ways in parks are by default for pedestrians unless signed). This is supposed to contrast with informal ways that are neither signed nor covered by any law in any way, for example, a footpath in the middle of wilderness that does not appear in official maps (usually mapped as highway=path). So, yes, footways in OSM do not require signage, but usually a little more formality than paths - not that most mappers and common people would reason about legalisms like that.
Here in Brazil, it’s quite a bit more informal though.