In my opinion, plain common sense dictates that the Imperium would have some sort of mechanism for day to day contact with the governments of member worlds and some mechanism for handling day to day problems of Imperial citizens on member worlds.
Hans
I don't see the Imperium getting involved in the day-to-day lives of most citizens at all.
Take CT4: Mercenary:
Traveller assumes a remote centralized government (referred to in this volume as
the Imperium), possessed of great industrial and technological might, but unable,
due to the sheer distances and travel times involved, to exert total control at all
levels everywhere within its star-spanning realm. On the frontiers, extensive home
rule provisions allow planetary populations to choose their own forms of government,
raise and maintain armed forces for local security, pass and enforce laws
governing local conduct, and regulate (within limits) commerce. ... Conflicting
local interests often settle thier [sic] differences by force of arms, with lmperial
forces looking quietly the other way, unable to effectively intervene as a police
force in any but the most wide-spread of conflicts without jeopardizing thier
primary mission of the defense of the realm. Only when local conflicts threaten
either the security or the economy of the area do lmperial forces take an active
hand, and then it is with speed and overwhelming force.