Do presynaptic neurons and postsynaptic neurons have different compositions of neurotransmitter receptors and transporters?
Typical bouton-spine synapses are formed as an interactive process between the pre- and post-synaptic cells (Scheiffele, 2003). There is communication between pre- and post-synaptic cell through guidance cues, growth factors, and structural proteins, as well as neurotransmitter release itself. Therefore, as part of the normal developmental process, postsynaptic cells will usually have receptors for the major neurotransmitter released by the presynaptic cells: they find each other rather than connecting at random.

Synapses that become inactive or have lower activity than their neighbors can also degrade and disappear entirely (Purves and Lichtman, 1980) by a competitive process (Balice-Gordon and Lichtman, 1994). Because of this process, I would expect that even if some developmental fluke led to a formation of a synapse between mismatched pre/post neurotransmitter release/receptors, the result would be that the synapse would get no stimulation and would be pruned according to the normal process.

However, some neurotransmitters can be released more broadly into the extracellular space rather than at highly specialized synapses (including serotonin, dopamine, and acetylcholine - note: these neurotransmitters can also be released at synapses) (De-Miguel and Trueta, 2005; Trueta and De-Miguel, 2012) and call also 'leak' from synapses into the extrasynaptic space: in these cases, the neurotransmitter is not likely to have an effect (or the same effect) on all nearby dendrites, only those that express fairly high-affinity receptors that can detect the neurotransmitter at an extrasynaptic concentration.

References:

Balice-Gordon, R. J., &amp; Lichtman, J. W. (1994). Long-term synapse loss induced by focal blockade of postsynaptlc receptors. Nature, 372(6506), 519.

De-Miguel, F. F., &amp; Trueta, C. (2005). Synaptic and extrasynaptic secretion of serotonin. Cellular and molecular neurobiology, 25(2), 297-312.

Purves, D., &amp; Lichtman, J. W. (1980). Elimination of synapses in the developing nervous system. Science, 210(4466), 153-157.

Scheiffele, P. (2003). Cell-cell signaling during synapse formation in the CNS. Annual review of neuroscience, 26(1), 485-508.

Trueta, C., &amp; De-Miguel, F. F. (2012). Extrasynaptic exocytosis and its mechanisms: a source of molecules mediating volume transmission in the nervous system. Frontiers in physiology, 3, 319.