Battles can occur with wild Denjuu, event Denjuu(Denjuu that challenge you when spoken to) or other T-Fangers.
Battle with other T-Fanger cannot be refused.
Winning a T-Fanger battle allows you to snatch the phone number of the last enemy Denjuu that fainted. Denjuu captured in this manner then consider the location you are at its home location - eg you got a phone number of an enemy Chamelan in Pepperi Mountain, when you call that Chamelan again in the same location it will arrive immediately like the Denjuu native to Pepperi Mountain. In this manner you can strategically capture powerful Denjuu to use in the subsequent battles or Denjuu to fill up the phonebook. There are some version specific denjuu that can be obtained in this manner.
The tournament in Machine Town Freesia is an exception to the above; you cannot snatch enemy Denjuu from other T-Fanger in the tournament. Also cannot snatch enemy Denjuu from Tabasco.
You cant take phone number of a defeated event Denjuu unless it gives it to you - eg you wont get the phone number of the Cortos you defeated to get back the toy.
Event Denjuu give up more exp and chiru than the same level wild denjuu, hence while you can walk around some of the event denjuu in dungeons, its better to fight them.
Only Ion Island when the phone companies are fighting the event denjuu will respawn, else defeated event denjuu just fades away.
Losing a battle causes blackout and losing half the chiru on hand and a random phone number from the phone[usually the most recently obtained], so keep extra phone numbers! And call the maximum allowable by the signal reception - dont call only 1 denjuu when you're allowed 2 and call the nearest ones.
The only time you are scripted to lose in the game is the 1st fight with Golaking at Cactos Ruins - your every attack will miss him, he can one shot your denjuu.