While there is extraordinary trekking potential in Sikkim and Arunachal Pradesh, strict government regulation and red tape makes it difficult, and illegal in some cases, to trek independently.
Sikkim and Darjeeling
To legally trek at higher altitudes in Sikkim you need a permit (obtainable through a travel agent) and a guide. Low altitude treks such as the "monastic loop" do not require permits and are easy to do independently. In addition to enter Sikkim a permit is also required, this permit is issued free of charge in Siliguri, Darjeeling and the Rangpo border post. The permit is valid for only 15 days but after that it may be renewed for another 15 days but you have to return to Gangtok to do it. This means if you do trek here you need to do some careful planning so you are not out in the middle of nowhere when your permit expires. There are only two routes that really qualify as Himalayan treks open in Sikkim both require additional permits. The permit for the Goecha La trek is easy to obtain assuming you go through a travel agent and have a guide, the other trek is the Green Lake trek which is a much more difficult permit to obtain. I wouldn't recommend trying to do it without a permit. If you have your own camping gear it may be possible to hike around the check point at the edge of Yuksom and do the Goecha La trek independently without a permit. I'm not sure what the chances of someone asking for a permit further up or turning you if they found you without a permit would be, but I would guess it would be pretty low. It is a beautiful trek especially in the spring when the rhododendrons are blooming. It's also the only trek I've done in the Himalayas with a guide, not because it was necessary to find the way (its pretty straightforward route finding), but rather for the legal reasons.
The Singalila Ridge Trek from Darjeeling used to be a popular easily independently done trek and then the powers that be took a page out of neighboring Sikkim's book and started requiring an unnecessary guide to do it. The whapping 350 INR a day for the mandatory guide has kept me away form this trek, although the views of the distant Himalayas including Mt. Everest are supposed to be quite good, weather permitting. I've heard of at least one person who talked his way through by telling them he did the trek before and didn't have a guide (when it wasn't required). I'm also not sure about the possibly of using alternate entry points and hiking around check points to avoid the guide. For me it was simpler to just trek somewhere else.
Best Season: April-May and September-November
Base Cities: Darjeeling (Shared Jeeps and Buses from Siligiri), Gangtok (Shared Jeeps and Buses from Siligiri), Siligiri (Train and Flights from Delhi or Calcutta)
Guide book/Maps: Lonely Planet Trekking in the Indian Himalaya
Arunachal Pradesh
Arunachal Pradesh has recently become much less restrictive than it used to be. It is now possible to get a month long permit to the state for 2 people for $100, where under the old rules you needed 4 people and you where only given 10 days. With a month long permit Arunachal becomes a much more feasible Trekking destination. You will need to bring all of your own camping gear and supplies and make sure you write down all the towns on your proposed route when you apply for the permit to the state. Both the Tawang and Menchuka regions appear to offer a great deal of trekking potential. The trick will be finding maps and routes, it may be worth enlisting the services of a local guide as there is very little information written about trekking in Arunachal.
Best Season: April-May and September-November
Base City: Tawang (Shared Jeeps and Buses from Tezpur or Guwahati, Assam)
Guide book/Maps: Lonely Planet Trekking in the Indian Himalaya
