First, your student must know the difference between these two articulations.
Usually beginner books have at least one song with staccato as an "amusement feature." This is usually something titled "Falling Raindrops." If not, write a song or two which includes staccato notes (both hands! Don't forget to develop both hands equally!) This one song won't be enough, so take one of those bland beginner method pieces and put in some staccato notes. Better yet, hand the pencil to the student!
Staccato is easy to play, actually. Many beginners do so naturally and must be trained to play legato. The trick at this point is calling forth staccato on demand.
Let's now assume the student can play staccato notes and legato notes in both hands. Now you're ready teach him to play one articulation in one hand and one in the other.
I like to use a 5-finger pattern on C (RH thumb on Middle C; LH 5 on small C, which is an octave below Middle C).
First, ask the student to "play one hand staccato and one hand legato at the same time." 10% or so of students can do it right off! Now you've saved yourself a lot of time! If so, ask the student to reverse hands.
Now the other 90% must learn this skill, and here's where you come in!
I like to describe it as follows, "You're going to play both notes at the same time [in this case, the first Cs], and one hand is going to come up while the other stays down. Try that. [Disaster, usually.] Ok, let's try it this way: you play both hands at the same time and I'm going to tap on the underside of the hand that is supposed to come up. [Tap on the palm.]" Usually this works just fine. As the student plays the note, you say, "Down." When you tap on the palm, you say, "Up!" Eventually you don't need to tap, but you should continue saying, "Down . . . up!" until the student doesn't need the "training wheels" anymore.
At this point in my studio, the student launches into a set of rhythms which helps him solidify this newly-acquired skill. After all rhythms are done in one hand, we reverse the process and learn staccato in the opposite hand, followed by rhythms.
An additional benefit of using rhythms as part of the technical regimen is that when the student gets to the point where she needs help in learning a note-y section, the rhythms are already well-forged into the hand and can be called forth to solve the problem at hand. I keep this a secret, though, and roll it out ("ta da!") at the appropriate time for the maximum effect.
I don't recommend that you teach both LH staccato/RH legato and LF legato/RH staccato at the same time. Let the student master one of them first.
To communicate this to students is sometimes difficult. I always mention the pattern training and the warm-up aspects to children. To adults, I mention the flexibility/strength issues because many adults worry that their hands are "too stiff" and/or "too weak and untrained" to allow them to be successful pianists. Technique, for adults, is a very logical solution to a problem which vexes them, and they usually embrace it willingly.
You may also be interested in the file on my technical regimen for students.
copyright 1998, Martha Beth Lewis, Ph.D.
Contact me for reprint permission.