Most consultancies have clauses in their employment contracts or employee handbooks that prohibit moonlighting. However, in reality, if you openly raise openly your self-employment and it is in a field that doesn't compete with your main job for clients, time or energy, then most firms will be sympathetic and will consider waiving this clause. Another factor is seniority - if you are a senior advisor then there is likely to be a lot more backing for you doing activities that build your network. If you're a fresh graduate then most firms would be less likely to trust you to manage your time between two jobs and to have the experience to manage conflicts of interest.
Needless to say, given how networked business is, it is absolutely the best idea to declare if you're operating your own business rather than trying to keep it quiet. It only takes one client from your independent business putting a "helpful" recommendation on your corporate LinkedIn profile to give the game away.