Yes, you should think about the implied harmony. Don't get hung up on the terminology of dyad, triad, chord. A basic definition of chord is three notes, but that is more properly a triad. A chord is just an abstract harmonic idea that doesn't need a specific number of tones. Your question deals with understanding two-part harmony where you can even have seventh chords with only two notes! Ex.
F above in
C is an incomplete but unmistakable
V7 chord! When analyzing two-part harmony you end up mentally filling in the missing, implied voices.
I like to think about the tendency of chord inversions to decide how incomplete chords could be understood and then the voice leading possibilities.
When you see a fourth the implication is a second inversion chord or a suspension. Either check to see if the fourth above moves like a suspension or fill in the missing third. Basically fourths get special handling so the options for the full harmony are a bit restricted in that sense.
A plain third is ambiguous. It could be the root and third of a root position chord, or it could be the third and fifth of a first inversion chord.
From those options try the possibilities.
Consider the implications for chord qualities when choosing the third voice and ask whether you are trying to hew to conventional harmony. In conventional harmony diminished chords often act as dominants or passing chords, and augmented chords are fairly rare.
Regarding conventional harmony think about voice leading and root progression. Two voices moving by step is roots by descending fifth, the "strong" progression. One voice moving by step is roots by descending third, a less strong but common progression. Three voices moving contrary to bass is root by descending/ascending step, not so common except for
IV V. Those aren't the only movements, just essential ones.
So, from your example: the first thing I thought was the fourths doesn't move like a suspensions so treat them like a second inversion chords. Don't worry about their qualities of major/minor yet. The important thing is to start with knowing their roots which are
Next, try harmonizing the thirds as either root position or first inversion chords. For these you don't really need to worry about chord quality, because you are adding either a perfect fourth or perfect fifth above the potential root and so the thirds of the potential chord will already be determined.
The two example below are how I completed the chords. I added the third voice in the treble the bottom bass clef is just to show the resulting chord roots...
The first uses roots by descending thirds and a harmonic sequence between the two bars. Descending thirds isn't the strongest progression, but the first and last chord are
Gm outlining a root relationship by fourth/fifth. There isn't a proper dominant, but it certainly outlines a basic
C minor tonality.
The second uses roots all by descending fifths and also the harmonic sequence. I used
E natural for the first chord, because then the two sequential pairs are dominant to tonic movements, like
Fm: V i | Eb: V I or
Eb: V/ii ii | V I.
As @LawrencePayne says there isn't one "right" way to complete the harmony. consider the options and take whatever suits your purpose. My first example could be a good pop progression where eschewing the tonic is common. The second is super conventional for classical style, so conventional it has a nickname, "the fonte."