Here is a listing of all the things that do and don't happen when the Filet/Net is used. Some of them may surprise you.
Who gets it:
The net can target a brute (including a reinforcement) or a pet. If there are any pets on the target side, it will target the pet. If there is more than one pet, the net has an equal chance of targeting any of them. If there is a reinforcement brute, but no pets, the net has an equal chance of targeting the reinforcement or the challenging brute.
If a pet is targeted, it is ensnared until the end of the match, and it can do nothing unless it is freed. Except in very rare circumstances (see below), the only thing that can free a pet from the net is a bombe. No opponents will attack the netted pet.
If a brute is targeted when a reinforcment is present (or vice versa), the brute in the net will not be attacked. All attacks will be directed against the non-netted brute until the reinforcement's time is up. When that point is reached, if the reinforcement is under the net, it will break out and leave the fight. If the challenging brute is under the net, the reinforcement will leave and the netted brute will be targeted by the very next attack.
When reinforcement brutes have a net:
When a reinforcement brute uses a net, it is just like when the challenging brute uses a net, unless one of them use a net while the other one's net is still trapping something. In that event, the second net will almost always trap the same thing that is already netted. Once in a very rare while, when there is a netted pet as well as a free pet, the two nets will strike different pets, and two pets will be netted. In that case, the reinforcement may attack the pet that is under the challenging brute's net, and the challenging brute may attack the pet under the reinforcement's net, thereby freeing the pet. In either case, the nets will both remain in place and neither pet will move until its net is broken.
What can free a brute from the net:
When a reinforcement is netted, it will break out when its time expires. Any attack against a netted brute will break the net. This includes the super attacks Bombe, Marteau-pilon/Hammer, Voleur/Thief, and Deluge. (Since super attacks never miss, this essentially wastes the effect of the net.) A netted brute will only be targeted for attack if there are no other targets (except in the case of bombe). In the case of Voleur/Thief, the net is broken during the weapon stealing portion of the super, and the attack will not automatically hit.
What happens when you use your net:
When you net a brute, assuming there is no reinforcment in the other side of the arena, the next attack against it will automatically hit. If there is a reinforcement or pet on your side, it is possible that they might deliver the next attack, but if not, you gain all the initiative. That means you can use potion or fierce or trappeur, or all three, without the possibility of losing your turn. Note that if you make a combo attack, only the first blow of the combo is guaranteed to hit. Perseverant/Persistent will not assist you with any of the hits in the combo!
What happens when you are under the net:
As stated before, your brute can take no action while under the net, and the first hit of a combo will automatically hit. If you have ballerines that hasn't yet been used, you will still be hit, but the ballerines will take effect on the next hit. The first hit cannot be blocked or dodged, but if you have tete-de-fer/iron head, there is still the same chance you will disarm your opponent. After the first hit in a combo, you may block or dodge any remaining hits. However, Contre/Counterattack will not give you an attack if you block, and Pugnace/Hostility will not give you an attack at the end of the combo.
When a net is used during match initiative:
When a net is used on a brute (not a pet), all initiative is reset to zero. All bonuses gained by Sang-chaud/First Attack are lost, and all penalties imposed by Meditation are removed. If the net targets a pet, initiative is not affected.