The Team Rocket Duo
A closer look at how Jessie and James work together throughout the Kanto and Orange Islands episodes reveals the following patterns:
Jessie keeps James on track because he often suffers from crippling fear (thus, his preference to follow rather than lead). With her leadership and decisive/controlling nature, his talents in impersonation and disguise are put to better use, giving him direction and purpose. Conversely, James reminds Jessie of her humanity and the need to look out for others. With his conscience and big heart/gullibility, he activates the nurturing side of her, which seemed lost after her failure as a Pokémon nurse. Their strengths and weaknesses are thus counterbalances for each other.
In battle, her aggression and his defense balance each other out--Jessie's Ekans (later Arbok) is usually the primary attacker, while James' Koffing (later Weezing) provides cover with a smoke screen effect. They seem happiest together when they're plotting a new scheme, and their most successful stages of attack are brainstorming and disguising themselves.
...and The Cons
However, neither of them are quite the consummate spies they pretend to be--the combination of James' anxiety and Jessie's arrogance almost always lead to failure in the end. They are frankly and hilariously terrible at being bad (but Jessie won't admit it, even when James shrewdly points it out). In some ways, they self-sabotage more than they are actually beaten by Ash & company; their plans always contain a glaring flaw that neither of them happened to think about. James usually slumps into despondence over their failures, but Jessie responds with flaring anger, often abusing James in the process. Sometimes, at the end of episodes, it looks as if they are the worst-put-together team in the history of anything!
Outside of Dastardly Deeds
There are glimpses here and there of Team Rocket's life outside of plotting and failing to capture rare Pokémon. In a few episodes, we see James preparing meals, or Jessie setting up and defending their various campsites (flipping the traditional gender dynamic). Other times, we get an extended look at one or more of them, such as in "Snow Way Out!" (Jessie's backstory), "Holy Matrimony" (James' backstory) and "Go West, Young Meowth" (Meowth's backstory).
In each of these cases, we learn more about Team Rocket as well-rounded beings, not merely mustache-twirling bad guys. They become characters who have to eat and sleep, who have experienced loss, absence, and heartbreak, who each have their own reasons for having turned to crime as a lifestyle. And sometimes, it seems they would be happier as a pair (or trio) if they just put all this Pokémon-capture nonsense aside. In ways, they are better than villains, but not quite heroes.
Can't Forget Meowth!
It's difficult to determine whether Meowth adds to or detracts from Jessie and James as a team. Sometimes he comes up with the best, most-thought-out ideas, and sometimes his selfishness and harsh criticism rivals and exceeds Jessie's, to the detriment of the team.
However, it's hard to talk about Jessie and James without bringing up Meowth. In a way, he is almost like a child to them (albeit a spoiled one), since he can speak human language and interact with them. Meowth's presence definitely distracts from the romantic aspects of Jessie and James' interactions; without him, those overtones become harder to ignore.
I find that Jessie and James are still very much a duo even with Meowth present. Jessie and Meowth may usually come up with the more ruthless schemes together, but Jessie still seems to need James' support and talents to pull them off. James often ends up rescuing Meowth, but he doesn't follow Meowth's lead--he follows Jessie's. Neither of them let Meowth dominate the team, which humanizes them in a pseudo-parental way.