Not that I'm good with radios, but I'll take a stab at this.
What the next stage is looking for is the difference in frequencies so I would guess the L.O must be of a rather high amplitude.
The lowest amplitude signal in the mixer will determine the amplitude to the next stage and you have some control over the L.O. so it seems you would be sure the L.O. amplitude isn't the weak link. Whatever your supply voltage is, I would guess you should make your L.O. voltage most of it.
For an "active" mixer (like the NE602 Gilbert Cell), it's like #12 said - the L.O. level depends on what the manufacturer specifies in the data sheet.
For a common diode-ring (e.g. double-balanced) mixer, the L.O. levels are typically specified between +4dBm and +13dBm depending on the model of mixer chosen. Each diode-ring Double-Balanced-Mixer (DBM) is designed for a single "optimum" L.O. level, typically +4dBm, +7dBm, +10dBm, or +13dBm. These mixers are often called Level-4 mixers, Level-7 mixers, etc.
Each L.O. "Level" mixer comes with trade-offs regarding L.O. level versus dynamic range, intercept-point, etc. A common DBM type is a Level-7 mixer. The Level-7 DBM has a good balance between required L.O. level, conversion loss, L.O. rejection, intercept-point, and other characteristics for general-purpose use.
Using the correct manufacturer-specified L.O. level is very important. If you don't, you will be compromising the mixer's performance and may even cause unwanted spurious products in your design.
I suggest you go to the site linked below and read all the application notes under the category "Frequency Mixer".
A mixer's conversion loss depends somewhat on the L.O. level. DBM's with L.O. levels between Level-4 an Level-13 will all have roughly 6dB of conversion loss, but DBMs outside this range vary quite a bit more, especially for L.O. levels below Level-4. Read the app notes a the link above for lots more.
If you want to make your own DBM using common diodes and a couple of toroid cores, try to get your L.O. level up to around +4dBm to +7dBm, especially if you are using silicon small-signal/switching diodes (e.g. 1N914, 1N4148, etc.) If you can use small-signal Schottky diodes, you may get away with a lower L.O. value like 0dBm to +4dBm.