This answer is offered as partially addressing the question, however imperfect this answer may be. The proposed question is very interesting but the attempt to answer is fraught with logical dilemmas. We are asked to assume the planet in question is essentially identical to earth in the advancement of its technological development, and then determine what particular aspect of this planet would be most prominent or obvious to us if observed at several light-years distance. Our means of observation is with an highly advanced observational probe that is located within the solar system, which for the purposes of this discussion, will be considered as parked in a relatively quiet environment deep within the Kuiper belt. We are to assume this observed planet is within a star system that is nearby, but given the characteristics of the planet being observed, we could well be observing earth with the query how would we recognize earth has intelligent life?
For this planet, we are asked -
...might we most easily notice patterns in its radio transmissions (or, if not) might we most easily notice bio-indicator chemicals in the atmosphere...
In general, the answer to these questions is no. But let's assume for a moment that we have been able to observe something telling about this planet. We are challenged to determine the implications of a) the question that is being asked, and of b) the evidence we have found.
In considering this question, we must be careful that we are not establishing the premise of our answer based on notions of things or aspects that are currently present in our everyday technological environment. The only exceptions might be principles of function. For instance, we cannot presume the people of this observed planet have a notion of the passage of time, even though they, as we, understand the presence of time. An interesting way to view the evidence would be to ask, for instance, about the people of this planet and what we could possibly know about them. If we take technology, for instance, all we could say is that if we hear their radio transmissions, they understand radio and have achieved the ability to develop technology. Regarding size of their technological elements, for instance, we could say the physical size of their computational platforms is likely not huge. Further, that these people are likely not huge. But nevertheless, neither can we say that any life form reaching this state of technological advancement has likely undergone a path of evolution parallel to ours even though parallel evolution would almost certainly lead to intelligent life and the development of technology. An error would be to think they look like us, but there may be inferred similarities. For instance, to reach a technologically advanced state, they would have evolved with an anatomy compatible with the manipulation of technological tools and elements, and the assembly of complex and wonderful technological devices. Those tasks require intelligence of some sort, and dexterity. Think about our ability to use a screw driver or pliers, for instance, or in an advanced state of technology, use of photo lithography in development and assembly of miniaturized, large-scale electronic circuits.
Considering the development of civilization and technology, the practical consideration is this: During the 6 billion year history of the planet earth, the window in time for development of our own nascent state of technology has been most exceedingly small, say, much less than just the last 250 years. And earth has undergone 6 mass extinctions in just the last half billion years of that 6 bln year history. Considering the chances of an encounter with a civilization at, or during, this short and most recent 250 year span, given the vastness of time over which this life form has evolved to a civilized state, the unknown age and state of evolution of this advanced life, the evident limitations and exceedingly small probable occurrence of such life on planets within habitable systems in the galaxy, and the expansively vast size of the galaxy, we can see that a chance encounter with another civilization within the time period of this nascent 250-year period of technological development is essentially and virtually non-existent both temporally and spatially. Consequently, the technological advancement of the supposed civilization we have detected may be assumed to be much older and far more advanced than ours, in fact much older by many tens of thousands or hundreds of thousands of years, or even millions of years, if not older. Because this is so, we would not recognize any technology they may have. We may detect the possibility of their presence by evidence within the chemical composition of their atmosphere, and if we detect their radio transmissions, given our total inability to understand their technology, we would find those radio transmissions exceedingly strange and heretofore unknown. Let's step back and consider this again, but from a different vantage.
So what exactly would we see with our advanced observational platform, looking at them from, say, 1.2 parsecs. Atmosphere: hydrogen, oxygen, some carbon, nitrogen. Composition: water, carbon-dioxide, trace methane. Radio transmissions, none, except as those possibly noted within the quiet zone in the vicinity of the 21-cm hydrogen-emission band. What is so special about 21-cm vicinal radio transmissions? For the most part, planetary atmospheres, and to a large extent, all other galactic radio noise, will block or obliterate radio emissions well below the hydrogen radio-emission wavelength of 21 cm. However, planetary atmospheres like ours, are transparent to, and galactic radio emissions are absent from, the quiet radio window surrounding the 21-cm hydrogen radio emission. Consequently, if this planet has an active civilization, and they use radio, we may detect some of their radio transmissions within the vicinity of the 21-cm hydrogen emission. In fact, we benefit from absence of noise for transmissions within this quiescent radio region. Think GPS, for instance.
What would these transmissions sound like? Well, of course, we have no idea. However, regarding radio transmissions, considering the state of our technology in radio, and assuming their state of technology in radio is similar to ours (even though more advanced), we may be able to hazard a guess regarding their uses for radio within the 21-cm band. How would we know what the state of their radio technology would be? Again, of course, we would have no idea, but the principles of radio have been sufficiently investigated and well understood over the time that radio technology has been developed on earth, that we can say our radio technology is as good as theirs except for innovative advancements they have which we have not considered or are unknown to us. Like the principles of physics, the principles of radio have been sufficiently explored and understood that our state of technology is sufficient to be equivalent to their state of technology in many ways, even though ours may seem archaic or primitive or ancient in comparison to theirs, and missing advanced technological elements. The point is this: If they incur radio emissions within a quiet area of the radio spectrum, then at 1.2 parsecs we will easily be able to hear them, otherwise if they are transmitting radio that we cannot hear, their radio transmissions are so soft they cannot be heard or they are transmitting within a noisy part of the radio spectrum. Consequently, without radio, or our ability to hear them, they would be undetectable. Our search would move on.
So here is the kicker: If we were to hear radio transmissions from this planet, how would we know what they are? Further, we will certainly find we have no way to understand what they are. All we could surmise would be that their radio transmissions are not frivolity, but rather are carrying real information of some sort, even though we cannot determine what that information would be. Keep in mind, if we cannot establish the premise of our observations based on notions of things or aspects currently present in our everyday technological environment, including language, then we would also not have the ability to understand these radio transmissions, even those from earth, if, in fact, earth was the planet we were observing.
For additional context, see my answer regarding the Fermi paradox, here.