I have never used two. But it is my guess that both are HRP conjugated than you may not distinguish between them. If they were different conjugated than might be possible
Why don't you use several blots and treat them independently? Using the same blot is not a good idea. Of course, it depends on what you use for revelation otherwise you won't be able to distinguish between the two of them.
Antibodies don't care (especially secondaries): they'll bind their target epitope whether other species antibodies are around or not. We routinely use multiple mouse primaries with a single anti-mouse secondary, and on occasion use both mouse and rabbit primaries with mouse and rabbit secondaries.
IgM vs IgG (of the same species) is slightly contentious, depending on cross reactivity, but in this instance even that doesn't matter: both are HRP conjugated, so you're at maximal cross-contamination already. You won't know whether your HRP signal is from antibody 1 or antibody 2, so you can ignore this as a factor.
The important questions are:
1) are your target proteins different sizes? They should be, and by a significant factor (like, 20kDa or more) -this helps make it very clear which band is which.
2) are your primary antibodies highly-specific (i.e. do they both ONLY produce a single band when used alone)? If they generate non-specific background, it becomes much, much more difficult to claim your two signals are legitimate.
3) are your secondary antibodies also highly specific? It would be worth running two blots, each with a single primary alone (IgG or IgM) but both probed with both secondaries (IgG and IgM), just so you can confidently claim that the IgM secondary doesn't pick up any IgG signal, and vice versa.
4) do they have compatible dilution factors? This will be key for HRP antibodies where (presumably) you'll be using ECL methods to develop. If one primary/secondary antibody pair is very high affinity, very high abundance, and they other is low affinity low abundance, you'll need to titrate the antibody concentrations (both primary and secondary) such that both produce comparable signal, or the first pair will burn through all your ECL and you'll never see the latter pair. For reference, we often co-probe for dystrophin (low abundance target, fairly rubbish antibody) and vinculin (higher abundance, really good antibody): while we use the same secondary antibody for both (anti mouse HRP, 1:50000), the primaries are used at 1:20 (dystrophin) and 1:100,000 (vinculin), purely because the signal disparity is so high otherwise.
Long story short, though: yes. There is nothing inherently wrong with this approach, as long as you address all the caveats above.
If you have two primary Abs from two different species (ex. mouse and rabbit), use their respective anti-species secondary Abs either conjugated with HRP (if both the proteins of ur interest are of not same size) or fluorochromes (not an issue even if they are of the same size as you can split the channels using ImageJ)