Interpretation of scattering parameter
I am assuming that since you say you are observing the S-Parameters you have access to a Vector Network Analyzer (VNA) or at least a Scalar Network Analyzer (SNA).

Your musings about S-parameters are not all correct. There are several things you should be aware of. When you're using the "a" and "b" coefficient convention instead of voltage ratios \$a_1\$ is almost always assumed to be equal to 1.0. For a passive device \$S_{11}\$ and \$S_{21}\$ are always also related by conservation of energy, i.e., they are not independent:
$$
|S_{11}|^2 + |S_{21}|^2 = 1.0
$$

This just states that all the power input to a passive 2-port device is either reflected or transmitted. This does not account for radiated power, or loss due to physical properties of your material / device.

You are also confounding the linear scale used in text-book derivations/descriptions of S-Parameters with the displayed values of your measured device, which is almost certainly in decibels (dB). We will address this in a few parts, along with other parts of your question.

Regarding your question about standing waves: in linear scale for \$|S_{11}| = 1.0\$ indicates that all the power is being reflected from the input and only standing waves exist on your transmission line. There is also another concept called the Reflection coefficient \$\Gamma\$ (which is equal to \$S_{11}\$ in many cases) from which you can calculate the Voltage Standing Wave Ratio (VSWR) of your device.

$$
VSWR = \frac{1 + |\Gamma|}{1 -|\Gamma|}
$$

Higher values calculated for VSWR indicate more standing waves (worse impedance matching) on your transmission line. All VNAs and most SNAs will be able to display this value as well. VSWR &lt; 2.0 is typically a value that indicates impedance matching that is "good enough".

On to insertion loss and return loss. For passive devices both of these values in log scale will be negative. Logarithms are very useful in RF engineering because they are useful for amplifying variations that are very small in linear scale to values that are easy to discuss in log scale.

Insertion loss shows you how much power is lost when passing through your device from port 1 to port 2 (values closer to 0 dB are better).

Return loss shows how much power is reflected back to the source from port 1 of your device (values in the 10-30 dB range are typical for devices that are considered well matched). Every 10 dB is an order of magnitude (10x) in linear scale.

I think now that we have some background information established we can talk about your measurements.


  In reality I measure negative values for both parameters. |S21| is nearly showing a flat line close to -1 and |S11| has an average value of -15dB with a dip going down to -30 dB @1.5 GHz.


This seems like a normal measurement for both \$S_{21}\$ and \$S_{11}\$ and that your device is best matched at 1.5 GHz. Lets translate -1 dB and -15 dB back to linear scale and talk about percentages since it seems you are more comfortable in linear scales.

$$
|S_{21}|(lin.) = 10^{S_{21}(dB)/20} = 0.891
$$
$$
|S_{11}|(lin.) = 10^{S_{11}(dB)/20} = 0.178
$$
$$
|S_{11}|^2 + |S_{21}|^2 = 0.825
$$

This indicates that 89.1% of the voltage wave is transmitted from port 1 to port 2 in of your device. 17.8% of the voltage wave is reflected. Equation 3 indicates that only 82.5% of the power input into your device is transmitted or reflected, this means that 17.5% is absorbed (lost as heat) or radiated into space.

Your last two inferences are incorrect, always make sure what type of scale you are working in. You can only consider percentages in linear scale, so if your network analyzer provides results in decibels you have to convert to linear scale to get percentages. Always annotate the units next to measurements so you don't get confused, this can be especially confusing when working with logarithmic scales because quantities (like S-Parameters) are unitless ratios, so always indicate that you're in log scale by putting dB next to your quantity.